In my entire career as a US software developer (15+ years) I've only had 1 employer that offered sick leave. Every other employer offered a "bucket" of time off. So even if you just have a cold you either go to work or you have one or two less vacation days that year. It's not uncommon to only have 10 days total for the "bucket", even for senior developers.
Well for starters, you typically have a limited number of vacation days a year, while sick days don’t have these limitations.
Secondly, in case of burnout (or similar), it’s not unheard of (at least over here in .nl) that you stay at home for extended periods of time, and gradually restart picking up your workload. That kind of thing is very different from vacation.
There are legal differences between sick leave and vacation, at least where I live. The former is required by state law, the latter isnt. Vacation hours can be accrued and have a cash-out value, sick leave will expire. There are a bunch of other rules as well.
You should never have to reveal the nature of any health conditions to an employer to take sick leave.