Yes 1961 was the last time they won the league, and the only other time 1951.
The season Leicester won it was an absolute freak, it was undoubtedly their one and only chance, so many all failed badly and all at the same time, it's like is extremely unlikely to ever be seen again, almost certainly not in most of our lifetimes.
If you actually examine who has knocked Spurs out of the cups, it is more often than not teams who you might expect should beat them more often than not, City, United, Chelsea, Arsenal. Yes Spurs would have a chance but they are hardly huge surprises. Don't get me wrong, a cup would get a monkey off his back but Spurs really shouldn't be competing with these clubs anyway, in financial terms they just aren't in the same ball park
To move Spurs to a position of regularly qualifying for the Champions league from formerly only spasmodically being able to do so, is a huge achievement given the restraints on them, they have moved from a team aspiring to be amongst the top few to actually being one of them.
That is massive, far greater than winning a reserves cup.
Being a Champions league regular will do far more to eventually get Spurs competitive, winning an FA or league cup, although nice, will do next to nothing.