Chelsea actually aren't a badly run club, it just seems that way because of all the in's & out's in terms of managers. Think I saw a stat somewhere that they've never went consecutive trophyless seasons since their first PL title in 04-05. Obviously you'd like managerial sustainability and maybe a bit of extra transfer market support for the manager coming off a title winning season, but it's hard to argue when their model consistently wins and challenges for trophies.
On your current debate, you're both right. Jose has little right to complain about team quality when he's had such backing, but he could've been backed a little further with Alderweireld tbf, but as you say, Alderweireld solves none of the problems on show last night, as good as he is. There's been a lot of embarrassment at this club since Sir Alex's retirement, and nobody is without some degree of blame for that, Sir Alex included.