You aren't firing a coach mid season to get better. You are doing it to rip the bandaid off and get a start on the next era now rather than later. This team isn't making the playoffs anymore. They are 1-4. The season is over if competing for a Super Bowl is part of the equation.
It achieves a few things. It allows the league to know we are serious about finding a new HC and means we are going to be able to begin to line up discussions and interviews early. It allows Kraft and the front office to stop having delusions about continuing with the current regime and take a look around the league and get a sense of what is working, what isn't, and what direction they might want to go. It allows them to look at the current coaching staff sans Bill and see if anything at all is worth keeping. If Mayo wants a future as head coach, this can be his audition for it instead of going into next season blind to what he can do and then having to decide if we want to give up on him as the successor or not. We at least get some intelligence on that decision. It allows Bill to not be a lame duck and if he wants to keep coaching he can be free to start looking early and line up his own job. Most importantly, the trade deadline is at the end of the month. Bill Belichick should not be in control of team management at that time if he's not staying on next year. Bill will not have the firesale we should be having. He will not be prepping the team for a ground up rebuild. He's not going to admit that type of failure or want to start over like that. If we are serious about going into the next era, he can't be part of the things we can be doing now to prepare for that.
You need to not think about about making decisions to help save this year. There is no saving this year. You are doing things to prepare for what is to come.