You’re absolutely right that it’s not the only distinction between wa and ga, but I think it’s the most relevant one here. Going into all the possible nuances and reasons for using one over the other is a bit much as a simple answer to a simple question, especially since not everything is relevant to the case presented.
I can’t personally stand Cure Dolly (it’s the voice… just can’t listen to it, no matter how good the content may be), but I can wholeheartedly recommend the explanation translated by our very own @Jonapedia:
I think the confusion around the emphasis explanation comes from what “emphasis” means in this sense. It’s not emphasis in the same sense as when you’d use bold text or extra stress in English, for instance - it’s just emphasis in terms of “this is the (more) important part of the sentence in terms of what I’m actually trying to convey”.
Small caveat: in many (I’d almost say nearly all, but I’m not sure about that) cases where it’s not the subject or object, you won’t see は on its own, but rather combined with another particle (には, とは, では, etc.) - but in the case of が or を, は (much like も) will essentially replace the case particle instead of being attached to it.
That’s actually a pretty important thing to realise about particles in general - they always mark some property of a word or clause, but that won’t always be the grammatical case of it. And that is indeed one important distinction: が is a case-marking particle, while は is not a case-marking particle, but can sometimes take on the role of one (が or を specifically).