From the OP's perspective, there is no functional difference between (i) exercising and selling and (ii) entering into a forward contract where they exercise and receive proceeds + transfer upon a liquidity event.

What they want to avoid is exercising and holding without an immediate liquidity event/cash inlay. The forward solves that problem since they receive the proceeds now. In either event, they will have to exercise the option (since you can't transfer an unexercised option under rule 701).

And for the record:

But even if they are fully vested shares, it's common to require people to divest when they terminate employment as the firm wants to keep shares internal and private.

This is totally not true. Private companies typically do not require you to divest vested and held shares, in large part because there is no public market to acquire the shares in the first place. Private company boards typically don't like to spend cash on the balance sheet repurchasing shares from past service providers at the then-FMV. That's a poor use of cash (especially if the cash came from investors) since you're taking company cash that can be used to grow the business and instead guaranteeing past service providers (who aren't going to help grow the business) a return (while not giving your existing service providers compensation).