This may be worth posting in the sports med forum? I assume you’re looking for research rather than anecdote
My non-research backed opinion would be that the reason you hear more about menstrual cycle vs contraceptive is because when you’re on the pill, you have muted hormonal changes throughout the month. Depending on which hormonal birth control you’re on you either have the same hormones all month (minus the period week) or much smaller changes throughout the month. That would lead me to conclude that it would have a similar, but muted, impact to the natural menstrual cycle. But I also think it probably depends a lot on which method you’re on as some methods have much higher hormone levels than others and I would guess that has a big impact.