S'cuse me, but "severe prudery" and "religious indoctrinated misogyny" are my business insofar as they affect our common socio-cultural hygiene. These ideologies do affect other people, so no it is not a private issue, especially not when it comes to public places. Not to mention that full-body burkinis are extremely unhygienic. Ask any Scandinavian why saunas are in the nude!
If you don't appreciate my culture (something that many generation before me have worked hard to build up, care for and improve on), don't come here. Burqas and burkinis make it pretty obvious that you have no respect for my western values. You can do whatever you want in your own private sphere, but such is not the case for the public sphere.
A burqa or burkini is not just a regular garment, it holds symbolic and ideologic power. How would you feel about somebody casually wearing a KKK hood or the famous shaved head & bomber jacket combo? These things have meaning.
Also has it never occurred to you how western women might feel very intimidated by these practices, knowing full well what misogynist belief structure is hiding behind it, enforced by men who expect women to dress up like that? As a woman I would feel very uncomfortable being confronted with this blatant disregard for my emancipative rights.
The thing about acceptance/non-acceptance all of you are talking about involves many things at a personal level and society level.
For me, I dont care about people wearing turbans or burkas etc.... because none of that involves me, and unless something really changes in my life, it wont matter to me. At work, there's some Muslim woman who rolls out a mini rug and prays at her cubicle a few times a day. She even pushes her chair our of the way into the aisle a bit for room so technically I got walk around it. Or I just avoid the aisle. Not the end of the world. Who cares.
I think the biggest issue with western countries isnt their usual PR image of "everyone is a buddy over here, c'mon over and do what you want", is it's that its easy to give, but hard to take away. So what happens is let's say theres some big issues popping up from religious stuff. Instead of telling that group to stop fucking around, the gov will be more like "Hey, locals, you got to learn and accept other people". The more practical and pragmatic thing to do is tell that group or belief to knock it off.
For example, a plant might require steel toe shoes, no loose clothes, no jewelry, hair tied back, maybe hair net too. Sounds logical to me, since if you give people the option to have loose things that may catch into a machine, that's giant liability issue begging to happen. If someone puts up a fight for a headscarf, that person may win due to discrimination laws due to religious beliefs. Makes no sense whatsoever, since a non religious person cant do the same.
Thats the problem govs face because most recognize religion and that gives people an out to do shit most people cant. If I was to jump into a community pool wearing jeans, the lifeguards would tell me to change clothes. Someone puts on a burkini, and it's A-ok. Makes no sense, but gov (who have no balls) doesn't want to get into heated matches leading to less votes, so they appease the masses instead something more logical.