...the multinationals are politically unpopular. They know the game is up.
Sorry Drew, but they don't and it isn't. And they don't really care about popularity, they care about profitability, which will always require the minimization of tax.
They'll pay more, sure, but they won't pay anything approaching the levels of tax expected of a purely domestic competitor who isn't finessing the system.
...the multinational ends up paying little to no tax because no country sees the full revenues or profits.
And that is why little to nothing will actually change. Making too much money? Increase the charges from your offshore supplier, that you happen to own.
If people actually knew the extent of what was possible with multiple companies in multiple jurisdictions, they'd insist their politicians and civil servants actually did some real work and sort it out (Lower, flatter, simpler taxes that are coordinated globally is the only way).
Faffing about with doubles and sandwiches is the very small tip of a very large and ever growing iceberg.
Just remember kids, tax evasion is illegal and possibly immoral, tax avoidance is neither of those things.