Asked some cis women who were on birth control but still got their periods about this once, and apparently the main answer was "I want to keep my period so that every month I have automatic confirmation that the birth control is working correctly and I'm not pregnant."
Which makes sense, especially if you're on the pill or the shot, because those methods are only about 93-98% effective based on actual use. (As opposed to ideal use, which tends to be about 98% effective.)
Actual use here means when the success rate is affected by things like missing a pill or two, taking a pill too late (they're supposed to be taken the same time every day), getting sick and throwing up too soon after you took it, etc. (Not sure off-hand what the corresponding factors for the shot are, besides reupping it late.)
And even with a 98+% efficacy rate, you can still occasionally get pregnant - after all, if something is effective 99% of the time, and you have sex 100 times, then statistically one of those times will likely get you pregnant.
Incidentally other forms of contraception, such as condoms, are only effective 73% to 87% of the time, depending on how correctly they're used. (87% the number quoted by the infographic, but I've seen varying numbers of the years.)
Which is why you should always use two forms of birth control!!
BUT NOT two of the SAME kind of birth control - if you use two condoms, they will often chafe against each other and rip!!
And regardless of whether you're on the pill/implant/shot/an IUD or not, you should always* also use a barrier method such as a condom or dental dam too, because NONE of the methods in the part of the infographic above will protect you against STIs!! (x, x, x)
*unless you are Completely Sure that your partner could not have caught an STI since their last test, generally because you're in a long-term monogamous relationship, neither of you is having sex with anyone else, and neither of you is living long-term with a chronic or recurrent STI
Still cannot fathom wanting to continue having my period, but if you're having the kind of sex that could get you pregnant, I can definitely understand wanting that confirmation, especially as abortion access is both increasingly restricted AND very much tied to how long you have been pregnant (e.g., a lot of people don't start showing until after it's illegal to get an abortion in their state, so missing periods is really the only way to know besides regularly taking pregnancy tests)