Turning over, but...

After a while in the shop, my bike ('81 250 CSR) is finally running! But only just.

It does not always respond when cranking the throttle, but it will idle off the choke just fine.

So far, I've cleaned-out the carb, new fuel lines and filter, oil change, new spark plug, good battery. What am I missing? Is this a timing issue / something else?

Sounds to me like this bike has sat for many years without being started and most likely you have a serious carb cleaning to do. You may have corrosion built up if bike was outside in the elements.
Assuming that you cleaned out the tank and carbs and bike has electronic ignition. Bike starts but doesn't run very well then I would bank on the possibility that carbs are still mucked up. When you hooked up to car battery? May have messed up something. Maybe not but at least you need a new battery, I wouldn't run battery on a charger after it has been dead awhile. You may not be running enough juice to make bike run properly. There should only be two things to matter with your bike, spark and carburation..

Could be a bad, or clogged petcock, ...is your tank rust free , & clean of any debris ?
are you getting a good spark ?

Spark is good, and I've cleaned out carb (maybe I should re-do?). I have a new battery that's totally charged (via battery tender, not car battery), so I'll give that a try and see if things improve.

For some history, this bike was rescued from a barn in WI., so it wasn't stored outside, but was subject to hot/cold swings. Before last weekend, it had been about a year since it was started-up last, so not a long-long time, but a good while.

I have not tried to let it idle/warm up and just run for 10-20 mins. Would that potentially help clear some stuff out?

Also, it's true that the petcock is old and needs some TLC, but the gas comes out just fine. I suppose I could to a tank clean and see if that helps the situation, but the fuel filter loos good, so I'm assuming I don't have too much gunk coming from that part of the system.