This demand contains an implied threat: that if you quit, you might expect a good reference, and if you are fired, you will get a bad reference. This is a very small consideration, especially if you have not seen the reference. The employee is asked to trade away several months of employment benefits for this unsure consideration.
The employer always wins if you quit, because their costs go down, and the unneeded worker is gone. The employee loses if their new company doesn’t check with their previous employer, if the previous employer gives a neutral or negative reference, if the previous employer folds and can’t provide any reference, or if the employee is unemployed for a long time and unable to get compensation because they quit “voluntarily.”
The employer knows if they badmouth the former employee and what they say is untruthful, it is lawsuit-bait. A larger employer is not very likely to actually say anything bad.
In the unlikely event that an employee thinks a good reference from this employer is critical to their future, they should get a letter of recommendation in writing, and read the letter to decide if it is worth the job they are giving up by volunteering to quit. They should also demand that a copy of the letter of recommendation be put in their personnel file, and be the agreed-upon reference that the employer gives.
There are some high level government jobs, typically political appointments who “serve at the pleasure…”, where a letter of resignation is customary. Unless you are being terminated as a cabinet secretary or attorney general, this doesn’t apply to you as an employee.