Why can't we have a lower league equivalent? Use some of the ridiculous funds made from the CL to spread the spirit of international co-operation and promote inter-european goodwill and so on and so forth.
Might make the little guys a bit of money too.
Oh wait, scratch that idea then.
This kind of happens with the Europa league, insofar as the early rounds start more or less as soon as the WC ends, and feature games from small UEFA members Wales, RoI, Luxembourg, etc.
If say, Aberystwyth Town qualify, and get drawn away to an Icelandic team, then I believe there is financial help from UEFA available for costs.
Of course it doesn't do much for lower league clubs in the bigger UEFA members, but the lack of interest in the Anglo-Italian cup when it was tried again in the 90's suggests, it would struggle.