If it becomes apparent that $CURRENT_EMPLOYER is not such a place, it's time to start protecting yourself with things like formal disability accommodations through HR, etc. The problem is that once this becomes apparent, shrewd politicians will have already worked to make it look like your accommodations are just a cover-up. That's not necessarily a deal killer, as every company puts a different weight on the legal risk that HR exists to prevent, but it's not likely to ingratiate your manager, which makes things difficult.
The long and short of it is that it's folly to imagine that HR is in your corner. They are paid by the company and they exist to keep the company free from the mind-boggling expense of lawsuits. You can work this to your advantage sometimes.
You sound like a manager who actually cares and empathizes with your employee. I had a manager like you. It was amazing. But you aren't the norm.
Going to HR will generally only draw negative attention. There is no situation I've seen where someone needed HR and HR actually did anything meaningful to support them.
I prefer to deal with each manager differently. Getting it in your HR file means you lose control over who knows what, and how it was explained to them.