First, there are some provisions where "late 50s" can be getting distributions, and 401(k) rules are different than IRA rules, and it matters if you still work for the employer or not. So, those are three things to read up on.

There is no reason to contribute to an employer retirement plan, just to have it distributed back to you. You mention there is another 401(k) at another job. Does someone monitor the big picture?

"Generally, you aggregate all elective deferrals you made to all plans in which you participate to determine if you have exceeded these limits."

Yes, you can go from 401(k) to IRA as rollover (meeting the timeliness requirement). If the payment went directly to his IRA account, there would be no fiddling with withholding and gross-up requirements. Because this: "adding the taxes withheld" is not Taxes Withheld. It's just Withholding, against any potential tax balance owed from 1040.

"and his income does not allow for a deductible IRA contribution in 2021"

"and Fidelity does not want to deal with this and just distributes the money (as a check, with federal and state taxes withheld) in December."

I'm pretty sure Fidelity cannot do this without his agreement, or at least, the plan administrator's. Fidelity cannot initiate this. I wonder if this isn't the action of a specific broker.