Jack,
What are you using to take the photos? Do you have a macro setting? Sweets should work fine, any cleaner is good. Going off the photos (which I recommend trying a macro setting for better pictures) the barrel does look very fouled. This can be fixed though.
How does the Mag look? On the ramp of the mag (assuming you're using OEM) can you see black or silver? You want this devoid of carbon. Do you have any local friends who you could borrow another mag from?
What type of .22 LR ammo are you trying to load?
If you don't mind, can you walk me through the process of what happened again? I know this seems repetitive, but we really need to narrow down the issue.
So you load your 10/22 OEM rotary mag with (insert the type of ammo). You make sure the mag is flush with the rifle and is fully inserted. You pull the slide all the way to the rear and let go (making sure to not ride the bolt home). This is where you're running into your issue correct? The Round will not fully leave the Mag and Chamber? Were you able to get any photos of this?
I wish I lived near you so I could help you out man. I've got tons of mags and different ammo. If you can, give us some more pics with the macro setting, show us where it's getting hung up, and tell us what ammo you're using. If you can walk us thorough my last paragraph, I think we can accurately diagnose your issue.
Best of luck until then!