I'm actually really curious for what the Pythagorean win percentage will say for this, so I'm going to run a hypothetical. For this assumption, I'm going to do the following:
1. To figure out a rough idea of how many runs a team full of Hayes would produce, I'm going to try to take an average of a few teams who are similar in OPS and average out to equal Hayes' OPS+ of 91. Coincidentally, if you average the total OPS+ of the Diamondbacks, Orioles, Royals, Angels, Marlins, Reds and Pirates, you get exactly a 91 OPS+. Those 7 teams have an average runs/game of 4.01, which is well below league average of 4.33. That would have a Hayes only offense be 24th in the MLB in runs/game and 27th in OPS+. Is this the most scientific way? No, but I don't know of a much better way to try and correlate an OPS+ to a runs/game
2. For defense, I'm going to go through the 2021-2022 seasons and take the best defensive player at every position and sum up their DRS, then average it out for the equivalent of 162 games (1458 innings). I don't think you can literally assign Hayes DRS to every position, I think a better analogy would be a team full of guys like Hayes, Simmons, Edman, Stallings and such than to just apply Hayes' DRS to all positions. I think using the "best" is also fair because Hayes and Arenado are clearly the two best defensive 3rd basemen. That gives you:
C: Trevino: 23 DRS in 1222 innings, adjusted to 27 DRS in 162 games
1B: Walker: 18 DRS in 1658 innings, adjusted to 16 DRS in 162 games
2B: Edman: 18 DRS in 1454 innings, adjusted to 18 DRS in 162 games
SS: Simmons: 19 DRS in 1195 innings, adjusted to 23 DRS in 162 games
3B: Hayes: 31 DRS in 1554 innings, adjusted to 29 DRS in 162 games
LF: O'Neill: 12 DRS in 1557 innings, adjusted to 11 DRS in 162 games
CF: Taylor: 27 DRS in 1796 innings, adjusted to 22 DRS in 162 games
RF: Tucker: 24 DRS in 1936 innings, adjusted to 18 DRS in 162 games
If you would have asked me who would be on this "all defense" team, I would have guessed Hayes, Taylor, Simmons and Edman for sure. Stallings was 2nd and would have been 1st had we not included this year, he was at 21 DRS in 892 innings but has been a huge negative this year (-5 DRS in 608 innings). If you sum up the DRS, you get a total DRS of 164 runs saved above average. My thought was to subtract those runs saved off of the average runs per game, so you're basically at -1 run against per game compared to the average because of that defense. In total, this leaves you:
Offensive: 4.01 runs/game
Defensive: 3.32 runs against/game
This team would be giving up runs like the Yankees and Dodgers, while only scoring them like the Mariners and Marlins. The closest real-life team I can find to these numbers (2nd best runs against but 24th runs for) was the 2016 Mets, who were 3rd in runs against but 26th in runs for. They went 87-75 and lost in the Wildcard. In terms of the Pythagorean Winning%, a team with that production would have 650 runs for and 538 runs against, which results in a .593 winning%. That's a record of 96-66.
TL,DR: With an average pitching staff, a team full of Hayes type of players would be expected to go 96-66 over a full season according to this.