Except they do already. The EUFA Cup winners go into the Champions League, so I don't think this issue is insurmountable. It would require a LOT of lobbying but why not give it a go? If you only offer it to countries with 3 or 4 Champions League places then I reckon Spain, Italy and Germany...
They could bring back the glamour of the FA Cup quite easily if they wanted to.
Semi-finals in neutral non-Wembley grounds
The final to be played at 3pm. No exceptions.
Bring back replays (but perhaps limit to one or two)
A nominated cheeky chappy from the winning side to be photographed with...