I worked for an organization in a salaried position that had a waiting period for getting health benefits at all. And it was itself a healthcare organization.
I also remember that you couldn't get FMLA leave before a certain amount of time passed. Which is why I left, because stuff happened shortly after I was hired and they wouldn't bend the rules.
There's no law that forbids you from giving someone time off without pay within that first year if you want to.
It's like when you get a privacy disclosure in the mail and it says "you can't limit the sharing of <some kind of information> because the law doesn't give you the right".
But you specified they wouldn't give you FMLA leave, not unpaid leave. If that's how you requested it, then they wouldn't be able to give you that, due to FMLA itself.
Well, my boss encouraged me to apply anyway. I am sure he was not so obtuse as not to understand what I needed, and when I gave notice it didn't come as a shock.
I also remember that you couldn't get FMLA leave before a certain amount of time passed. Which is why I left, because stuff happened shortly after I was hired and they wouldn't bend the rules.