How do I stop being asked to turn on two-factor authentication?

You can't stop that message. Two factor is the way things are going for anything requiring any degree of security. Since many people now use their devices for a lot more than just calling grandma on Sundays, many organizations are implementing it.

There is no inherent disadvantage to using 2FA with non-Apple equipment. Many organizations that are not Apple related are using two factor. I had to set it up it when creating a Gmail account and another service provider I was using was starting to implement it. In fact with them they required you to authenticate weekly. With Apple you only really need to do it once per item you use unless you are constantly changing your equipment. You may find the supposed problems are more related to things such as automatic cookie deletion or other features inherent to the system you are using.

Also remember you can (and should) have multiple ways to get a code, including a recovery contact. The people who really get in a mess with two factor are those who have an iPhone as their only way of getting a code and then that phone is lost or stolen.