So according to your logic winning the EL made Spurs a bigger club than Real M., Barca, Man City, Bayern M. because those clubs failed to win the CL.
Sevilla won the Europe League in 2014. 2015, 2016. Are they regarded as a top club?
If Real M. had reached 15 consecutive CL finals I am sure they would win a few of them

And they would have been regarded as a big club.
Anyway, try tell señor Florentino Pérez that instead of aiming to win the CL every season, something he will never be able to do, he should instead have Real M. aim for the EL because it's much easier to win. And if he does win the EL a few times his club will be seen as a giant. Tell him to lower his standards and win "lesser" trophies.
Why did the big leagues push to have more teams - on behalf of "smaller" leagues - in the CL knowing it would be harder to win? Do your really think they believe it's a bigger achievement to win the EL?
And why would Spurs fans want their team to finish in the top-4, even without a chance to win the EPL title, knowing that ending 5 and would give them an easier shoot at a title as it would qualify them for the easier EL?
It's trophy vs. prestige. And a trophy doesn't always equate beating prestige.