An out-of-the-box idea.
Since soccer nowadays is an entertainment industry with the goal of making money and pleasing audiences, why not let audiences pick up what they really want to see?
UEFA would generate revenues from televote and audiences would be pleased at their will.
With some constraints (keeping country protection up to some point, and seeding, only 1 vote per number) it could work. UEFA could make available a list of possible clashes and let televoters decide.
2) With a small twist (voters from one country could not vote for teams from that country), we could add this system also to decide PSO at the end of even ties. The principle would be: since we have to pick one, let the "neutral" Europeans decide which of the two teams "deserve" more to be in the next stage.
I'm assuming, as in eurovision dynamics, that with one-vote-per-number and a huge amount of voters from all over the continent a "reasonable" consensus would emerge and sympathies and frauds would be diluted in the mass of expressed votes.
Let televote chose the draw
Let televote chose the draw
Play fair and square!