Scrap international breaks during the season completely. Cut the number of games, and move the remaining ones to a single part of the calendar where all of the qualifiers are played. The obvious time would be June/July in odd-numbered years, but scrapping international breaks would significantly shorten the time taken to complete domestic seasons, so there might be other options.
I think longer-term there is a case for one big mid-season winter break for internationals could be useful with the rest played in the summer. It would ensure AFCON and the Asia Cup can be held outside of the peak of summer whilst also ensuring teams with African and Asian players do not see their players miss club games. It would allow international managers more time to work with their players - major tournaments are almost always the only time they actually play and train together for several weeks at a time.
It would also stop multiple international breaks during the club season as you highlight - there is one scheduled for September, October and November which does seem a bit over-kill. Its perhaps a secondary benefit but it would also mean less travelling. If you are Salah you go to Egypt in the summer and winter break travel around Africa to play competitive qualifiers for AFCON/World Cup or play AFCON and then return to the UK at end of summer and winter break rather than doing it every other month.
I guess one problem could be 'tradition'. For the 'product' you probably don't want 30+ PL games played before an International break so you don't want winter break too late but if you have it too early you skip out days like the 3rd Round of FA Cup.
A calendar something like this perhaps could be possible (presuming same start date as this season) - looking from perspective of PL - could be modified as required for other leagues (it should be easier in nations with 1 cup comp)
August: 4 PL games (14th, 18th, 21st, 28th), Carabao Cup 2nd round (25th)
September: 6 PL games (1st, 4th,11th, 15th,18th, 25th), Carabao Cup 3rd and 4th round (8th, 29th), Champions League (22nd)
October: 4 PL games (2nd, 9th, 16th, 30th), Carabao Cup QF (23rd), Champions League (6th, 20th)
November: 4 PL games (6th, 13th, 20th, 27th), Carabao Cup SF legs (10th, 24th), Champions League (3rd, 17th)
December: 6 PL games (4th, 11th, 14th, 18th, 26th, 28th), Carabao Cup Final (22nd), Champions League (1st)
January: 1 PL game (1st), FA Cup 3rd Round (8th)* - AFCON and Asia Cup players potentially miss FA Cup 3rd round
Winter Break: 8th January to 12th February - For AFCON and AFC Asia Cup. Other federations can play a few of their qualifying matches here.
February: 2 PL games (12th, 26th), Champions League last-16 dates (16th, 23rd), FA Cup 4th Round (19th)
March: 4 PL games (12th, 16th, 19th, 23rd), FA Cup 5th Round and QF (5th, 26th), Champions League Last-16 dates (2nd, 9th), Champions League QF date - 1st leg (30th)
April: 5 PL games (2nd, 9th, 13th, 23rd, 30th), Champions League QF date 2nd leg (6th), Champions League SF (20th, 27th), FA Cup SF (16th)
May: 2 PL games (4th, 14th) FA Cup Final (7th), Champions League Final (21st)
International Break starts 23rd May - nations have from 23rd May to 2nd July - either to play Euros/Copa America or play qualifying matches.
2nd July to 1st August - holiday for International players.
1st August - International players return training for start of 2022/23 season on 13th August.
It means most midweeks see football but for the international players that isn't much of a change. The Carabao Cup would likely also see players rotated in earlier rounds anyway. International players at least get a clearly defined break each summer regardless of Euros. Non-international players get a mid-season break as well as an end-of-season break.
It would mean the Championship especially (and L1 and L2 would see their fixtures re-jigged) but The Championship would have more free dates during the season outside of 1 big winter block.
For Africa and Asia with winter tournaments that gives them a bit more time in the summer to get in qualifying games for World Cup and qualifying for AFCON/Asia Cup. In the case of Asia it may be needed as 18 games minimum are needed to qualify for World Cup (20 if you are a 'minnow' entering at first round). Perhaps the format could be re-formatted (there are 2 group stages currently).
I've worked on assumption season starts on date it did this year but perhaps you could push the start of season later into August/September and push end of season back to ensure that on return of winter break you don't have any clubs that have little to play for after winter break.