Things are going poorly.
The game at Southampton exhibited a lot of the weaknesses of what this club is currently like playing under Jose. Southampton hadn't had a clean sheet in a year, but they had no problem keeping Tottenham off the board. I don't know how much Mourinho has learned in his time away from the game, but I don't see his strategy leading to winning football with this club, for a number of reasons.
* The defense is still bad, even with Jose coming in and slowing things down with two-holding midfielders. The backline has a lot of issues, Jan Vertonghen is playing LB as a way to get Sanchez and Toby both into the starting eleven. It does create a scenario where their three best defenders are on the field at the same time, but now Vertonghen is playing out of position. I love SuperJan, but they are asking a lot out of him. The sudden disappearance of Danny Rose hasn't helped.
* I don't know if he is truly as bad as he seems, but Aurier just is a complete disaster to me. He doesn't defend particularly well and he just so wasteful with the ball. Occasionally he can deliver a good ball into the box and he can run, but other than that I don't see much of a player and he feels like a big negative when he is out there.
* Gazzaniga has been pretty bad. Haven't heard much on Hugo's return, but the last bit of info seemed like he was targeting a January return so hopefully soon? They need him badly.
* Hopefully, one day, the two holding midfielder strategy will work when Sissoko and Ndombele are healthy. Ndombele has been great when he has been healthy, but has been constantly injured since arriving. He was bossing around the midfield during the first 30 minutes against Southampton, but then he had to come off. Sissoko has been the most consistent player all year, which is crazy when you think about where he was when he first arrived. Dier isn't starting player anymore and Jose clearly doesn't rate Winks too highly.
* Eriksen has undoubtedly been better since Jose took over, but still looks like a guy ready to exit the club and not a world class playmaker. Dele has played better as well but still looks extremely frustrated with the entire situation and also looks like someone who may need a change of scenery.
* Without Son, Lucas was being asked to play out of position and his impact was limited. When Kane went down, he slid into a more natural position and suddenly had a much bigger impact on the game.
* Haven't heard about how badly Kane is injured, but it looked like it could have been a torn hamstring. Needless to say the team will need to find another way to score goals without him, at least Son is back and Lamela looks fit.
* The new signings have been disappointing. Ndombele has been good but hurt, Lo Celso has been hurt and unimpressive when fit, Sessegnon didn't play, but looked lost when he was out there yesterday.
I agree with your overall assessment that things are going poorly, but I disagree about a couple of things.
The backline is bad, but I think it's looked that way in large part because the MF is such a mess. The club really needs a cleanup side-to-side DMF to cover the gaps behind the FBs and protect the CBs (a/k/a a destroyer). The silver lining here is that Spurs have a plethora of guys who could all serve as reasonable partners for such a player, thus the DMF doesn't need to have lots of skill or technical ability. Young Mascherano, De Jong (minus the reckless stupidity), Luis Gustavo, Gattuso, Medel, etc. are all good comps. Obviously I don't think Spurs is about to pip an all-timer like some of those guys, but since so many megaclubs want technicians at that spot (Rodri, Busi, Casemiro, Fabinho, etc.) it's a role that Spurs should be able to get a player for <20M. The real value is that it allows all the other mids to slide into their best spots (more on this below).
I really want Aurier to succeed, but he objectively hasn't. He's so inconsistent and he drifts often. It's sad because he has a nice skillset and I like his style, but the next few months are hopefully his last with the club.
I can't say enough good things about Sissoko. He's been consistent at several different roles and he's probably my player of the season so far (Dele's start was bad and Son's cards are becoming a concern). In one of the recent matches he was bombing forward from a deep position and I sometimes forget how good he is at creating numerical advantages. Getting a true DMF next to him would let him create havoc.
Ndombele has had flashes and he is clearly technically solid, but I was expecting more from him. Same goes for GLC, but I think the injury and lack of PT are reasonable explanations there.
Jose tried Sess at LB and I think most agree that is his best position for the future. I hope he sticks there, because the Jan experiment is not going as well as last year.
Kane is the man, but the system is so discombobulated right now that some of his best moments are when he drops between the lines and serves as the creator. Part of me thinks this squad will improve with a speedy Son and/or Lucas ahead of Dele. To your point about Lucas, what is his natural position? I see him first and foremost as a winger or withdrawn striker, but as I've said before I think he is amazingly skilled but struggles to fit into any system tidily.
Lastly, I think Gaz has been very good for Spurs. He's certainly had a couple of howlers and the last man tackles were not good, but he's far from the culprit IMO. One of the biggest questions that remains for me (both short and long term) is whether the CBs and the defense generally have looked so bad because of the MF, because the FBs, or because their own general drop in quality. Likely a combo of the 3, but then why are we signing Toby to an extension?