This does not work.

The spell Find Steed allows you to cast a spell and have it also affect your steed.

While mounted on your steed, you can make any spell you cast that
targets only you also target your steed.

In the targets section of the PHB, it states:

A spell's description tells you whether the spell targets creatures,
objects, or a point of origin for an area effect. (PHB, pg 204)

In the case of the spell Find Familiar, it’s description denotes that the target is also an unoccupied space within range:

Appearing in an unoccupied space within range

So the ruling that makes the target self spells transferable,

that targets only you

does not enable you to cast Find Familiar on your steed.

find familiar (likely) does not target the caster, and so it only summons one familiar

Under the Targets section it states:

A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area of effect...

And under the Range section it states:

The target of a spell must be within the spell's range...

Looking at find familiar we see that it has a range of 10 feet and that it says:

You gain the service of a familiar... Appearing in an unoccupied space within range...

Thus find familiar targets the space within 10 feet.

We also see under the "Targeting Yourself" section that:

If a spell targets a creature of your choice, you can choose yourself...

find familiar does not target a creature of your choice so you cannot choose to have it target yourself using this method either.

And furthermore under the "Range" section we also see that:

Other spells, such as the shield spell, affect only you. These spells have a range of self.

find familiar does not have a range of self and so cannot target you for that reason as well.

The target of the spell is the point in space, not the caster and so it cannot also target the mount.

Even if you wanted to say that find familiar targets you because it affects you by granting you a familiar in your service you still could not do this.
find steed states:

you can make any spell you cast that targets only you also target your steed.

However, find familiar also targets the point in space and thus cannot benefit from find steed's clause.

Summoning spells do not have a sole target of the spellcaster

As has been discussed to exhaustion on this site, both in posts and in chat, the nature of a "target", with respect to spellcasting in 5th edition D&D, is a very finicky and fragile concept that, in general, should not be receiving a lot of time being thought about.

So I can certainly understand how one might argue, starting from the premise that the "effect" of Find Familiar is to affix a bond between the spellcaster and a magically constructed companion, that the "target" of the spell is indeed the spellcaster, and the companion merely the "effect"; after all, the spell Fire Bolt "targets" a single creature and constructs a mote of Fire that is then flung at the target; one wouldn't argue that the spell Fire Bolt targets both the recipient and the mote itself, yes? Otherwise it would invalidate its use with things like Twinned Spell!

But in the case of Find Familiar (and, indeed, all Summoning spells), that isn't how we understand them. For starters, Find Familiar doesn't really imply, strongly or weakly, that the spirit bound to you was created wholecloth; the spell isn't Create Familiar, it's Find Familiar.

You gain the service of a familiar, a spirit that takes an animal form you choose: [...] Appearing in an unoccupied space within range, the familiar has the statistics of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast.

—Find Familiar, Player's Handbook, pg. 240

The implication being that the spirit existed prior to the casting of the spell.

So by any commonsense reading of the spell, we must recognize that the spell's targets must be described in one of two ways:

• The spell "targets" the spirit that becomes your familiar


• The spell "targets" the spirit that becomes your familiar and you, in establishing the bond between the two of you

Meanwhile, if we assume that the spell is, indeed, "creating" the spirit for use, well... It's still a spirit. An independent entity. Any commonsense reading of the spell must assert that the spell is indeed targeting the spirit; if not for the summoning itself, then for the ongoing effects on the spirit conferred by the spell.

Whichever is true, Find Familiar is an invalid spell to benefit from the "Spell Sharing" effects of Find Steed

Regardless of which version is true, this means that the spell does not target "only you", which is the requirement of the ability to "share" spells with the steed.

So Steeds cannot gain familiars through this method.

"But what if they could?"

Alright, let's ignore that first part. Let's suppose, for the sake of argument, that Find Familiar is valid for the spell sharing feature of Find Steed.

In that case, because the steed is benefiting from its own version of the spell, the second Familiar would be subservient directly to the steed, not to the steed's master. The steed would be able to control their familiar in all the same ways that you are able to control your Familiar.

Designer's Intent

You point out, yourself:

The spell indicates that you can’t have more than one familiar at a time...

This is Rules as Written, as well as a clear indication at the designers intent. This is clearly not within the intent of the designers, even if you disagree with the why it doesn't work bellow.

Some DMs may reward the cleverness, others may see it as rule lawyering rightfully (from a RAW and Rules As Inteded point of view) dismiss the attempt. It is clear the designers want to limit people to one familiar from the find familiar spell:

You can't have more than one familiar at a time.

Note that it isn't tied to a source, or exception. You simply can't have more than one. That means for balance purposes the game designers decided you shouldn't have more than one. I mean, from a sheer action economy, you'd get the steed's turn, your turn, and two familiar's turns -- while everyone else sits and waits.

There is an Exception

The spell flock of familiars is a seperate spell that contains an exception.

You temporarily summon three familiars – spirits that take animal forms of your choice. Each familiar uses the same rules and options for a familiar conjured by the Find Familiar spell. All the familiars conjured by this spell must be the same type of creature (celestials, fey, or fiends; your choice). If you already have a familiar conjured by the find familiar spell or similar means, then one fewer familiars are conjured by this spell.

D&D is based on the specific beats general. Note that the specific exception for more familiars is clear, and what the new limitations (1 hour duration, not a ritual, etc.) and changes to the base find familiar spell are layout. Nothing in find steed allows the casting of find familiars.

Moreover, this is in a module, which makes this new spell optional. The DM doesn't have to incorporate them into their campaign even if they playing the module.

Targets

The target doesn't have to be a creature:

A spell's description tells you whether the spell targets creatures, objects, or a point of origin for an area effect.

Find familiar doesn't target you at all. For instance no one would argue that fireball targets the finger of the caster despite reading:

A bright streak flashes from your pointing finger to a point you choose within range

We can all agree that it is the point in space that is the target. And magic missile isn't the caster despite beginning with the word "You". The target in this case is again, the people where the action happens:

You create three glowing darts of magical force.

The spell find familiar reads similiar to magical missile and fireball in this regard. You do something, and something happens to or at a target. You have not changed, the world has by the addition of the new creature:

You gain the service of a familiar, a spirit that takes an animal form you choose: bat, cat, crab, frog (toad), hawk, lizard, octopus, owl, poisonous snake, fish (quipper), rat, raven, sea horse, spider, or weasel. Appearing in an unoccupied space within range...

Leaving either the summoned animal form of the familiar or the space it appears in as the target (or both). Both are where the action of the spell actually takes place.

No Secret Rule

In a number of places you state "no secret rules" in D&D and I agree. Everything in this answer is quoting the source material all players and DMs have. Nothing is hidden. The target being the space where the summoning happens or the animal form that appears are both called out similar to other targets in other spells. What is also clear is that this spell has no exception to the one familiar rule.

Steed's Familiar

Alright, your DM has ignored all the rules on targets and rewarded your idea with a second familiar. Who controls it?

As you are claiming you are target of the spell, then all the you's of the "second casting" would refer to the new target (the steed). Som, all the you's would be replaced by the steed:

Your [The steeds] familiar acts independently of you [the steed], but it always obeys your [the steeds] commands.

In that case the DM would be in control (act's independently) but it would follow the steed's (who is under the DMs control [act's independently], but following your character's commands) commands to the best of its ability.