The M2 Carbine was put into production in April 1945. The war in Europe ended in May of 1945. The US also made conversion kits, so those may have made it earlier. Either way, it was pretty much over for the Germans by the time the M2 arrived.
Logistics is the biggest problem in non-standard weapons. The Germans captured large stocks of Soviet equipment early on, so they had a local logistics support system they could use. They even went to the trouble of rebarreling Ppsh-41's to 9mm in order to ease logistics as they ran through the captured ammo.
Remember they also continued production of weapons in countries they occupied. e.g. Nazi marked Browning HP or Norwegian .45 autos. The logistic concerns still existed, but were less since they would obviously continue to produce the ammo alongside the weapons, and in many cases were calibers they already used 8x57, 9mm, etc.
Look at our own use of the AK early in OIF. We had captured sufficient stocks of weapons and ammo and put them to use because we needed them. The Germans had the same situation and used whatever they could lay their hands on.
There were far less US stocks of supplies captured by the Germans on the Western front. Sure, there's "battlefield pick up", but you didn't get whole units issued substitute weapons like the Eastern front.
The carbine was likely popular for the same reason it was popular with the US. It was light and small. A guy that has to lug a Panzerchreck is going to appreciate having something better than a pistol and lighter than a rifle. An NCO/Officer (the same original audience for the M1 Carbine in US use) should be doing more directing his troops than shooting, and a light, small, carbine fits the bill. Espeically if it frees up one more rifle that can go to a shooter, and with the fewer number of rounds a leader will shoot, then the logistics issues aren't as important.
The big place for captured weapons was second and third line troops. POW camp guards, interior guards, police, etc. would have far less of the logistical issues because they wouldn't be in direct combat and need resupply as much. I've seen photos of POW camp guards armed with all sorts of odd-ball stuff.