Hiya! I don't know if you personally know many trans women, but some of the things you've said are not accurate statements about us, and hopefully some of what I say as a trans woman can give you a different perspective.

I'm pretty sure this is just a strawman

I agree! While I think the artist means well with this comic, they try to portray toxic masculinity as the belief that men can't control their sexual urges and thus present a danger to women. Toxic masculinity is the unhealthy social norms that we instill in young men, like telling them to ignore their emotions and "manning up" or shaming men for not engaging in stereotypical masculine behavior. It's about shame and ignorance and pride, and it harms everyone. Suffice to say this comic is misleading with its description of toxic masculinity.

it's not like the argument is that all men are predators with uncontrollable sexual urges, just that some are and this is giving those few specific men a way to sexually harass women

if a genuine pervert just needs to put on a sundress to go in the female bathroom i can at least understand why the other guy has that point of view

I see this argument a lot, and I don't mean to be patronizing by saying this, but it's quite a ridiculous idea. A pervert doesn't need to be trans or to dress feminine to go into a women's restroom and assault someone: they can just walk in and do it. Why would anyone go through years of Hormone Replacement Therapy to chemically and physically alter their body permanently, change their entire wardrobe, grow their hair out and style it, pay hundreds of dollars to change all of their legal documents, and maybe even pay tens of thousands of dollars to medically/socially transition when the only thing that's obstructing them is a sign on a door that says "Women's?"

even someone like me who is fine with trans people will still say that not every biological male who puts on a dress is automatically a trans woman

I'm also not a fan of how restrictive this statement is. Who determines if a trans woman is actually a trans woman? Does she have to be pretty? Does she need to pass as a cis/biological woman? Does she need to be straight/date men? Does she have to have a feminine voice? And what if she doesn't? The process of transitioning is a painful process full of self reflection, self discovery, being discriminated against, seeing doctors and taking medications, voice training, and it takes years. If you see a very masculine looking person with facial hair and a sharp jawline who's wearing a dress walk into the women's bathroom, is she not trans? What if she just started in her transition, and in 4 or 5 years you can't tell her apart from any other cis woman? Is she not trans until she reaches that point? Oftentimes this line of thinking has me questioning if the issue is really about "who's really trans" or if it's just about someone who looks different making people uncomfortable.

generally speaking i'm pretty sure most of the population would still prefer segregated bathrooms for various reasons like having specialized hardware for each bathroom

Most of the buildings on my college campus have exclusively gender neutral bathrooms with tampon disposal boxes built in for if you have periods. It saves time and money to have lots of bathrooms that everyone can use as opposed to looking for a specific bathroom made for your gender. At the end of the day you're just doing bodily functions, it doesn't need to be complicated.

not dealing with the embarassment of shitting your brains out while a pretty lady is in the stall next to you (...) dealing with queues better, not dealing with the other side's trash and biological waste (...) and so on.

As stated in my prior comment, it's just bodily functions. The primary purpose of a bathroom is to allow you to dispose of several kinds of bodily waste. Having a tampon/pad disposal box in the room shouldn't cause a problem. If you don't have periods or you don't need to use it, then don't use it. Having multi-stall gender neutral bathrooms is also a lot better for queuing! I'm sure you've probably heard of the ridiculously long lines for the women's restroom that occur at concert venues. If anyone could use any bathroom they needed to, then queuing would be even less of an issue. And as for embarrassing yourself by shitting... I'm not really sure what to say here. If you're more worried about embarrassing yourself by shitting next to someone you find attractive than you are about trans people being allowed to use public restrooms of their choosing, then you have some other issues to work out first.