How big is your budgie cage?

It's the Petco white finch flight cage.

That's a good idea. The playgym can be their "home away from home" where they can hang out when they're out of the cage. Put something that they love there, like millet spray.

And, I have noticed that my birds are most active in the morning, so maybe if I try to get them out of the cage more, I should do it when they're most willing to move.

That's a good plan! I don't know if budgies are the same, but cockatiels tend to be more cooperative in the evening so you could give them a try then too and see what happens. Morning and late afternoon/evening are the main foraging times in the wild so you'll be working with their natural instincts at those times. Midday tends to be more of a "hang out in the trees doing nothing" time.