1) I don't think a target man works in our system. Ade's problem this season wasn't just his attitude, it was that he was just way too static (the same issue Balleteli and Lambert have in Liverpool's system). Kane is so right for us, because he's so mobile (as Ings is).
2) I wouldn't bother with a 3rd CF unless they could also be our second LWF (Rodriguez) or were a teenager. 3 senior players into 1 position doesn't really go)
1) I know I've asked before, but I just don't get what you're basing this on.
Lambert worked really well under Pochettino at Southampton. When Poch was at Southampton the only striker they signed was Osvaldo, perhaps not a true target man, but he's at least as target man-ish as he's mobile. Part of the reason Lambert hasn't worked out at Liverpool is that he simply hasn't been given enough time on the pitch. In fact I would be rather happy if we could snap up Lambert come the summer, him and Ings as replacements for Soldado and Adebayor would be great transfer business imo.
Pochettino picked Adebayor consistently until it just didn't work (seemingly not just because of the pitch performances). He picked Adebayor over Soldado ever so consistently. Soldado has essentially never been given a run by Poch, probably because he's not seen as a good fit.
Kane is mobile, but he's also certainly fairly big and strong. Not Benteke-esque strong, but a powerful player capable of battling with centre halves at least.
I see no evidence at all that Pochettino doesn't want or doesn't use a target man option. I also see no reason to think that his system doesn't work better with someone a bit big and strong up front. Aren't we looking to emulate Dortmund and Athletico at least to an extent? Lewandowski, Costa, Mandzukic. All great players, all more than just old fashioned target men, but all big and strong players capable of really battling with centre halves. In fact Lewandowski dominating the Real backline physically was one of the standout features when Dortmund beat Madrid. Simeone immediately replacing one big strong striker with another, when Klopp lost Lewandowski he signed Adrian Ramos and quite possibly the lack of physicality up after losing Lewandowski front as Ramos hasn't really worked out has been part of their problem this season.
2) It wouldn't be 3 senior players for 1 position though. You yourself say that both can play as a #9 and as an AM. And that Ings can play as a LWF. How does that add up to 2 players for one position? It's almost like saying that Lamela, Eriksen and Pritchard would be 3 senior players into 1 position because all of them can play as the central #10. Another striker in addition to Ings would give us 3 options for the lone striker position as well as plenty of options for the attacking midfield positions whilst keeping the door open for a 2 striker solution and making sure that we would have a striker on the bench even when someone was injured.