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I think something that's not clicking for a lot of people on the left with the whole "Tampon Tim" thing (beyond general misogyny and transphobia) is that in conservative areas, there can actually be a pretty huge stigma on tampons vs pads
At least some areas where I grew up in the South, there were even some parents that would outright refuse to let their daughters use tampons (especially in middle school/junior high)
I remember going to the pool and seeing some girls just uncomfortably sitting off to the side, because they were on their period and were only allowed pads. Then in general, there were times people would offer each other products during an emergency, with some girls apologizing and awkwardly refusing tampons even when that was the only thing available, or girls apologizing they couldn't offer any tampons because they weren't allowed to have them
Even though for the most part there is a logical acknowledgement that it's just a hygiene product, there's still the general phrase I'd hear of "Nothing should go up there until marriage"
Then even beyond that, periods in general were seen as something personal and embarrassing that you were supposed to hide however you could (and this is still true in most places). Even a married man with children could get mocked for buying his wife any period products
With all of this, Tim Walz crossed multiple lines by specifically and publicly making period products available - especially tampons, especially in middle schools, and especially in boys' bathrooms. Boys (and people in general) shouldn't have to be aware of periods, and especially not something as morally questionable as a tampon.
So yeah. For some people "Tampon Tim" really is a deeply cutting and uncomfortable insult, to associate him with something (seen as) disgusting and borderline immoral that no one should ever be associated with.