Xjyuk

"Je découvre votre site et m'inscris dans la foulée car séduite par la bienveillance et la pertinence des commentaires lus dans plusieurs discussions."

I'm surprised to see that "car" is not followed by "je suis" and directly by "séduite".

This an ellipsis of subject and verb found commonly enough. It is found with other common grammatical words, for instance "parce que"; the tense of the ellipted verb can be dubious in some cases, the context is then needed to know what tense is correct;

• La machine s'est arrêtée parce que (elle est) dépourvue d'une lubrification essentielle. (could be "elle était")


• 
  Bien qu'(ils soient) ignorants des conventions, ils essayent de se familiariser avec ces pratiques.

In strict settings (formal and literary usage), car is a coordinating conjunction that links two parts that belong to the same category. It could be either:

• clauses: Je m'en vais car il se fait tard.
  


• or adjectives: Appelée longtemps « méthode champenoise » car utilisée en Champagne, cette méthode produit […].

Therefore, your sentence should normally be written as you rightfully expected:

Je découvre votre site et m'inscris dans la foulée car je suis séduite par la bienveillance et la pertinence des commentaires lus dans plusieurs discussions.

In casual speech car isn't used so much (parce que is preferred), but if it's ever used, I would certainly not expect an ellipsis as the sentence would become even more unnatural. I suppose you may find ellipses in written “telegraphic style” in very informal settings (comments on the internet could very well be of this type). But if you're used to English usage, be aware that omitting je suis in French sentences such as the one you provided won't be considered correct, and that they won't be spoken in such a way.

I don't really know the name of this rule but, to avoid repetitions you can omit the subject.

Il chante, il danse et il rit.

Can be written :

Il chante, danse et rit.

Reprise ou omission du pronom personnel sujet

"car + adj " is accepted by TLFr. This is a good reason to consider this usage as correct ( though the quotations given by TLFr could be discussed).

As far as I am concerned ( as a native french speaker, without any special authority however) , I tend to think that :

"parce que séduite par ... " or " étant séduite par ..." would sound much better

For example, one can say : " Jean n'est pas venu à la fête , parce que malade". Or, better, " il n'est pas venu, étant malade". But " il n'est pas venu, car malade" really sounds strange and infelicitous.

Below a link to a philosophy paper written by a reknowed academic with 2 occurrences of " parce que + adjective"

https://www.cairn.info/revue-cites-2015-1-page-73.html

You may search the expressions " parce que femme" , "parce que handicapée", " parce que riche" on google.

So : (1) no doubt about the correctness of " parce que + adj" (2) serious personal doubts as to the correctness in literary/ classical french language as to the correctness of " car + adjective".