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

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.

Short answer :
(1) the sentence you refer to would certainly be understood

(2) it would not be used in oral speech, for it ( mistakenly) aims at literary elegance ( which makes it precisely ridiculous)

(3) it is clearly not grammatical since the coordination conjunction ” car” does not allow an elliptic construction ( though some other coodination conjunctions do allow this construction)

(4) the friendliest advice that can be given to a person who leans french is : in case you have to produce literary / academic writings in french , avoid carefully this construction

Grévisse, Le bon usage ( generally considered as the essential reference regarding grammatical correctness in french) rejects this elliptic usage of “car”. The deep reason is not precisely that this usage is elliptic ( for some coordination conjunction can be constructed in this way) but the fact that, originally, the non-elliptic sentence ” je m’inscris car je suis séduite” is not rigorously correct.

Here is what Grevisse says, page 1400 ( https://archive.org/details/LeBonUsagefrenchpdf.com/page/n1400)

(1) “car” only has a semantic similarity with ” parce que” , ” puisque”. This semantic similarity is limited;though : ” the sentence ( or sub-sentence) the word ” car” introduces expresses not the real cause of the state of affairs previously referred to, but the justification of what has been stated “. So ” the word ” car” cannot introduce a sentence answering the question “why?” “

• Je minscris
• Pourquoi?
• Car je suis séduite par la bienveillance —> NOT CORRECT!

Since ” je m’inscris, car je suis séduite” is already not correct, the corresponding elliptic construction ( “je m’inscris, car séduite…’) is a fortiori not correct.

(2) from a syntactic point of view: using ” parce que ” in an elliptic construction is correct, using “car” in an elliptic construction is not correct

Here is what says Grévisse : ” on account of its semantic similarity with parce que and puisque, “car” is sometimes abusively used in the following sentence-construction : ” il parlait du nez, car enrhumé”.

Conclusion : ” je m’inscris car séduite” is abusive , in the same way as ” je m’inscris car je suis séduite par la bienveillance… et que…” . The origin of the mistake is a semantic similarity of ” car” with ” parce que ” which make poeple think that ” car ” can be used as a conjunction of subordination ( while it is a conjunction of coordination) in the same kind of construction as ” parce que”.

In another place of the same book, Grévisse is less severe and says that this usage is ” a little risky/chancy” see §268, R4 ( Remarque 4) – Grévisse gives quotations, none of them being from classical writers.

Note : what precedes does not imply that elliptic constructions are not allowed in general for conjunctions of coordination ( See Grévissse, §268)

Some explanations on Grevisse”s statement : the word “car” ” cannot introduce a sentence answering the question “why?”.

To understand this, one has to come back to a traditional philosophical distinction between ” ratio essendi” and ” ratio cognoscendi“.

The ” ratio essendi” ( raison d’être) of the climate change is its cause,that is the fact that the quantity of greenhouse gases in the atmosphere increases. Here the order is the real ( causal) one : observed effet + parce que + cause

So I can say : ” le climat se réchauffe parce que la quantité de gaz à effets de serre s’accroît”.

In the same way ” Non, ce n’est pas sa voix normale. Il parle du nez, parce qu’il est enrhumé”.

The “ratio cognoscendi” is the reason not of the fact itself, but of the fact we know it, and, more broadly, of the fact we assert a certain state of affair.

The function of ” car” is to express this “logical” or “epistemic” relation a sentence has with its justification, or between the asssertion of a sentence and the justification of the fact that it is asserted.

So here, the order is reversed : alledged cause + car + observed effect

• ” Il est enrhumé, car il parle du nez” ( the reason that justifies my assertion that ” il est enrhumé” is the observed fact that ” il parle du nez”).

• ” Le climat change, car les glaciers fondent.” ( what proves that ” le climat change ” is the observed fact that ” les glaciers fondent”)

• ” elle s’intéresse à ce site, car elle s’y est inscrite”