I have the same apprehension, but anime is a medium. There's a lot of disgusting books and Western movies around, and we don't shame people for liking books and Western movies.
The problem with anime is that some of the most popular ones suffer from the issues described in the meme. Take death note for example, immensely popular. But the female characters are mere props with one massively problematic female character's infatuation with the lead driving half the show. It's just unwatchable if you don't like that kind of thing.
Everyone has a comfort level and/or tolerance on what they can watch. Generally speaking, they're free to look away from content that they're uncomfortable, like the aforementioned example of Death Note with how the women are portrayed there. I have my own categories of content that I'm too uncomfortable to watch, and I'm sure it's different from what you find uncomfortable.
Again, it's not exclusive to anime. Personally, I find James Bond movies' (massively popular with quite the cultural cache) portrayals of women more problematic if applied to real life, but it's a work of fiction and we're supposed to know better. My comfort level does not preclude me from watching those as well as Death Note, nor I'm supposed to assume watchers and fans of those endorse all the themes involved in there.
Some critical thinking from us should be employed as we watch those things especially if sensitive themes are included. Where does it lie in the prescriptive vs. descriptive writing spectrum? Is it meant to teach the audience that something is good or bad (the should/shouldn't)? Does it describe a particular situation within the internally-consistent rules of the in-universe world (the what, where, when, how, and why)? etc. This applies regardless of the medium.
True. To me, it's the popular perception of anime that's the problem. Most people associate it with the op's topics, even if that's not everything. There's some critical point where there's enough weird stuff that it gets rounded up into "all anime is weird".
I honestly think it's just a good reason to always qualify your preferences. There are dozens of anime for which, if the conversation turned to that anime specifically, or even a major theme, I'd happily talk about it. But most of it really is complete trash. Honestly you can say that about most media which isn't "Christopher Nolan films."
Also, that's not to say complete trash can't be entertaining or have some marginal value. But anime does have a serious problem with sexualizing children, and that definitely keeps it at arms length for me as a general topic.
I qualify mine by saying I like giant robots. That's not 100% restricted to anime (and neither are my interests) but leaves the door open to discuss anime shows.