It's a thought, but if football went down this road, I wonder if it would be better for it to be similar to players... so clubs can sign a free agent at any time, but staff under contract with other teams only during transfer windows. Sacking stays unrestricted, but options for replacements are limited.
So a team could sack their manager and appoint someone unemployed (or perhaps an assistant manager or a coach at another club, though we could forbid that too) but couldn't poach someone else's manager outside the transfer window. Or during the season, with a manager window open only in the summer.
Think it's the poaching of someone else's manager mid-season (or unsettling rumours) that's the main disruptor, and that can be very unfair on the poachee's team.