Would sending voyager have been a real definite deadline?

Visiting this many planets was only possible due to a very rare alignment. It's a once a century event. That's why we sent two probes, not just one

Absolutely. You could wait decades or centuries for a useful planetary alignment.

Not really. Jupiter alone is good enough. Its huge mass accounts for almost all of the gain you get from any such slingshot. Launch windows from Jupiter to anywhere occur every 12 years. Voyager's alignment was captivating, but realistically if it hadn't happened, we would have just done separate Jupiter-Uranus and Jupiter-Neptune missions instead.