I'm not sure that Tottenham entered the season with a particularly small squad. Looking at Transfermarkt, I count 9 senior defenders, 6 senior midfielders, and 6 senior attackers, then add in a few academy players like Moore, Lankshear, and Scarlett.
Liverpool has 8 senior defenders, 7 senior midfielders, and 6 senior forwards. Arsenal entered the year with 9 senior defenders, 5 senior midfielders, and 6 senior attackers, plus Nwaneri and Lewis-Skelly. Take out the youth players who are in the Moore/Lankshear bracket and United have 10 senior defenders, 6 senior midfielders, and 6 senior attackers. City entered the season with 7 senior defenders, 7 senior midfielders, and 6 senior attackers. Chelsea is a crazy club, not even going to count what is going on with them.
Basically, all these sides went into the season with 20-22 senior outfield players and a few academy players as "break glass in emergency" options.
Squads have stayed more or less the same size for the last 10 years for a variety of reasons (squad size rules, managers not wanting unhappy players on the bench, money in the game leveling off, etc) but the fixture list keeps getting busier and the game keeps getting more intense. A run of injuries is really a killer in this context, as it puts so much load on the other players. Ange should be doing better and deserves the sack IMO, but I don't think its really a squad construction issue so much as a combination of bad luck, his tactical style maybe being conducive to causing injuries, and his tactical style definitely not being conducive to coping with an injury crisis.