In a few videos on the internet I've seen people get stuck casings and then immediately try to pull them out with a screwdriver or needle nose pliers.  Has anyone here ever seen this technique work, or do you pretty much always have to tap it out with a cleaning rod?

Just depends on how tight it is stuck. I've had about a 60% success rate prying them out with a screwdriver against the rim or needle nose the few times I have been caught without a rod.

pogo method works, but I also use heavy extractor springs

if not a semi=tap with rod

Depends on the rifle. If the case is buried in a barrel extension like an AR, just get a rod. If you have access to the rim like on an AK a screwdriver can work.

AJ

Normally, I try to abstain from running rods and shit through the muzzles of my rifles.  Here's a few methods, outside of running a rod down the bore, for some popular rifles:

You talking AR?  Mortar the son of a bitch, unless you are running any buttstock made by CAA, FAB Defense or LuthAR.  You will destroy those stocks.  Ask me how I know.    When mortaring an AR, if you have a case/round that is super stuck, I found it easy to tie a string/rope (550 cord is best, shoelace in a pinch would be good too) in a loop, wrap it around the CH and pull on it while mortaring the rifle.  It allows you to get more leverage on the gun.

You talking bolt?  Unlock the bolt and take a rubber mallet to the bolt handle.  If no rubber mallet, get a piece of wood as a buffer, place it against the bolt handle, and pound it open.  You can also apply the mortar method on a bolt gun too.  Obviously, your pressure is just applied to the bolt handle as opposed to the CH of your AR.

You talking AK?  Unpossible!  Rounds/cases never get stuck in AK's.  LOL.  Mortar it.  Stick a damn stainless steel or cast iron dowel, right through the muzzle, into the barrel, and beat the shit out of it with a sledgehammer.  I'm kidding.