I think taking sick days for mental health is exactly what it's for.
And within some reasonable amount, it's not unreasonable to take it after a really awful night of sleep or crazy 18 hour shift the day before, etc.
With white collar office work we can do remotely, we've all been lulled into taking essentially 0 sick days truly off. I've probably taken an average of 0-1 per year in my career. Most of the time any of us are sick we, at most, offer that we'll be offline a couple hours and login for the rest of the day.
Our parents generation had jobs where they accrued 5-7 days/year of sick day, and actually rolled it over like vacation days.. sometimes with no limits, allowing them to retire a year early paid, etc.
We've let this all get chipped away to the point that many of us don't even have vacation days that rollover, or if they do, must be used within Q1 and then get zeroed out on March 31st.
Never mind a fixed allotment of sick days we can feel comfortable using without suspicion.