No, they initially got it from the NEIGHBOR'S rubbish. They'd been watching him go over and put stuff in the neighbor's trash.
The officers were tasked with tracking Kohberger so they could arrest him as soon as a warrant was issued as well as trying to get hold of a DNA sample.
www.dailymail.co.uk
If in fact he was putting his bio-material in baggies first (or even just a garbage bag), he made it easier for them to find it. He took the time to remove his trash from his parents' property and relocate it elsewhere, but apparently did not look around to see if there was possible surveillance. Perhaps they were using a night drone. Perhaps there was a good place for a car to park, unseen (but how then does it have a line of sight to the Kohberger's house?)
I figure they probably just watched him leave the house in the wee hours of the morning, carrying something, and coming back without it. Common sense would tell investigators that he could be trying to hide something - my mind goes immediately to trash, since that's what LE looks for when they want DNA.
All those pairs of gloves have DNA in them and for them to be effective in preventing more transfer, they need to be changed often - all of the gloves being a rich source of epithelial DNA on their insides. So he has to throw them away and he doesn't want to put them in his own trash. IIRC, it was more than a few yards to the neighbors' house, so he would have been seen walking up that street to the other house(s). He may have used more than one neighbors' trash, we don't know that.
I assume he cleaned and recleaned both his car and apartment, although the car poses some problems (did he drive it to a self-serve car wash in some place that's not Pullman and not Moscow? Because he had to have been worried about being caught on camera cleaning his car. Maybe he took the car out east of Moscow to clean it on the plains. No one knows, but I assume he did his level best to clean it. DNA is hard to destroy, though.
IMO.