Looking at that article where he says 'pregnant people' it just sounds bonkers. It's such an unnatural phrase. If he'd said something like 'it's understandable that when you're pregnant you may be anxious about taking any medication or having vaccines .." it would have been much less contrived and performative, but still inclusive of anyone who gets upset at being reminded that they're female (though pregnancy is one hell of a reminder in my experience!).
If he's genuinely concerned about not upsetting someone with a male reproductive system who doesn't identify as a man as well as getting an important public health message across he would have done more than just parrot the 'people with prostates' line. It makes it sound like a health condition in itself, not an organ that all (or virtually all) males have.
I'm also puzzled about how he thinks you'd know whether you had one or not if you're male. If you'd had investigations for another issue (infertility? urinary tract issues?) you might, and in many cases it's poor function that's the issue so a man would still have a prostate and be at risk. I also know someone who had his prostate removed in his late 60s and was diagnosed with prostate cancer again in his mid 80s. He'd assumed it wouldn't be possible because he didn't have a prostate, but apparently it is.
I don't think Dr Xand thought this through at all. Probably thought it's more important to use wokespeak than to be clear and truthful.