I get what you're saying, totally. And it is perfectly acceptable to disagree with aspects of a "cause" while still supporting the people involved.
But (hah) that's not really the point of the comic.
It's when you feel the need to throw in your "but" after a statement of support that is at issue. It's not the fact of being free to have objections to the pharmaceutical industry using animals that's the problem in your example. It would be, in the context of this post, that you felt that your objection was tied to the right of trans people to exist with equal rights.
See, objection is to a completely separate issue. You can focus on reducing or eliminating the use of animals in medicine without tying it to the right of trans people to have access to gender affirming care. If your objection is to the animal issue, great, wonderful. That's a separate conversation.
And I'm only using that example because it's the one you used. It could be anything, any "but".
Like, my little issue that intersects enough that it could be a but is allowing dedicated spaces. There are times and places where a branch of cis gendered experience is being shared, and someone that isn't part of that group is going to be extraneous or possibly disruptive. Like, you don't have a support group for men that survive testicular cancer and think it's okay for a cis woman to show up. It's okay to exclude women from that. It's also okay to exclude anyone that hasn't had testicular cancer, even if they have testicles. It's also okay to exclude people that never had testicles at all, even if they're men, regardless of being cis or trans.
That, however, is absolutely unnecessary to bring up when I say "I support trans people". There's no but there. The support is full stop. No buts. There's zero need to drag that separate issue that just happens to intersect in a peripheral way with some segment of the trans population.
That's what the comic is about, not blindly accepting things. What it comes down to is that if you think your "buts" are more important than supporting the rights of trans people, you aren't really supporting them. And that's what adding that "but" means. It's saying that whatever your issue is is more important, that it overrides that struggle.
That's it. The presence of the but in that statement indicates it isn't true. And that would be the case for any "I support" statement.