In terms of what you *could* do with a bluebird, it probably wouldn't help you much, except in certain circumstances, like...

Suppose you have a balance on card #1 about to start accumulating interest and card #2 with an introductory 0% APR offer that also earns you >1% in rewards when you buy VRs. Instead of doing a balance transfer and potentially having to pay a fee to "move" a balance to a card where you wouldn't have to pay any interest, you could buy VRs where available (though it sounds like they're not) with a card that has a 0% purchase APR offer (at a cost of ~0.8% in VR purchase fees, so not "free" unless you value the rewards more), load the VRs to BB, and then pay off the card with an interesting-accruing balance with the funds from the VRs. You've basically just moved some debt around without necessarily needing to have the cash in your checking account first, and paid a fee to do so. But even then, you can only load $5K of VR onto each BB account each calendar month.

A second scenario could be that I have one card with a balance about to start earning interest, and 10 rewards cards, each of which offers rewards that valued more than the ~0.8% transaction fees associated with VRs. If timed properly (which may not be easy), you could use VR purchases on card #2 to pay off card #1 before it starts accruing interest. And so on down the line, as long as you don't pay interest and earn rewards that are worth more to you than the ~0.8% in fees for buying the VRs. With only one BB account, unfortunately, you'd probably need to start about a week before your payment is due to load $1k/day in VRs and then pay off a balance that's about to come due on another account.

But assuming you do have the cash to pay off your credit cards and don't have the ability to earn much interest in your checking/savings accounts, neither are better than just paying off your cards from a checking account, buying VRs on a rewards-earning second card, and then using BB to pay off the card you used to buy VRs.

For most people most of the time, it's easier to just pay off your cards in full every month. If you need/want to take advantage of 0% purchase APRs (or potentially even just run up a balance you're going to roll into a no-fee 0% balance transfer), it can have some limited added value if you pay very close attention to timing.