Absolutely nothing is preventing you from celebrating your birthday the way we do Holidays. My birthday isn't September 22, it's the third Saturday in September.
Heck it's not even weird to not bother with this redefinition and just have a party on a more convenient day.
There's nothing preventing that, sure, but I don't think I'd really enjoy that. I want to celebrate my birthday on the anniversary of the day I was born, not on the nearest weekend day.
(My birthday falls on a national holiday where I live, though, so in practice I've never had to worry about school or work on my birthday.)
In this calendar system, though, the third Saturday in September would always be the same day of the week, so it would keep you from celebrating in that format.
Nah, the opposite is true for me. If I know for sure it's always a Tue, I can plan around it. I can schedule PTO for Mon and Tue, and have a nice long weekend, for example.
What's so frustrating to me is when I realize that, for example, public holiday falls on the weekend (I know this is solved by some countries by moving the holiday to Monday or Friday). Last year, 24th of Dec was a Saturday, pretty much the worst time for Christmas.
That's not a reason to prefer one calendar over another. Celebrating the anniversary of your birth is an arbitrary decision. Preferring it to fall on a weekend is odf. Anyone can celebrate anything at any time.
Sure, celebrations can still be held on weekends, but it’s just not the same. And Fri 13th would die away.