That's the corrupting power of CL money, isn't. I mean, the FA Cup is a real prize that you can celebrate and remember and put in a cabinet and all that. It becomes part of the club's history.
None of that applies to qualifying for the CL, which is most likely followed by going out relatively early without any chance of winning it (for a club like Leicester at this stage of their development), and thus basically comes down to a non-first finish followed by a meh European campaign. But CL qualification now by many valued higher than the domestic cups because of the boatload of money that CL participation brings.
This is logical and unavoidable of course, because football is a big-money business, where success correlates significantly with success. But it does make football so much more boring...