Powered minecarts pushing paused carts

Solution 1:

In every attempt I've made at using powered carts, they've always come up short in terms of ability to push loads, not stick on things or travel far distances.

It's possible that changing the pushing device from a powered cart to nearby booster track would solve the sticking problem. There are many, many booster configurations that might fit the particular problem you're trying to solve.

Solution 2:

Powered carts simply don't push at all when they are not on track, and they don't have the 'excess momentum' that boosters create, so a powered cart will never be able to cross a pressure plate or other track gap by itself.