Oh come on - remove, repair and refit tube takes five mins maximum when you've got the knack..... (less if you replace the tube only at the side of a road). I did my son's front last night, and they are not quick release. That even includes re-fitting the tyre as my son decided he'd remove the tyre from the rim whilst the glue was drying.
I'd let the glue dry for a minute or two, and I don't usually bother to remove the plastic off the patch, unless it's had some time to dry - if a quick repair - the tube just goes back in tyre.
PS don't use cheap puncture kits - TipTop are the best, or any that use the round or oval patches with the black centres and the orange sticky edges.
Put your gears in the last but one (smallest) sprocket. Undo brake calliper - just remove the 'noodle' (bent tube) from the v brake (if fitted). Release QR lever, pull rear mech backwards, drop wheel out.
I'd recommend practicing it.
Let us know how you get on, as I suspect there may be some rough edge on the rim or the rim tape isn't seated properly causing the failure.
Don't bother about solid tyres - just don't work - keeping the tyres at the correct pressure is all that's needed.