I see your point, and I think that for us the issue is that the club and the fanbase has somehow decided that these competitions are beneath us. It started when Poch was here. The league and the CL, or getting into it, was all that mattered. And that's despite a few deep cup runs. Instead it was just seen as "ego stroking".
Look at how happy the Frankfurt, Rangers and Roma fans were last night to make their respective finals. These may not be the biggest competitions anymore, but I would love to win a few.
If we had a Europa League or a cup or two from the Poch years, Kane may not have pushed as hard to leave as he did last summer, and Poch would probably have been given more time to handle the painful rebuild.
But the counter argument will always be that going deep in those competitions means risking the league form. So you're damned if you do, and damned if you don't.