That includes using Scotch-Brite pads to clean valves or pipes, too. Or any sort of soap with "microbeads" or corrosive elements like bleach or the like. Anything at all. Clean water, PLAIN SOAP (like bottom shelf Ajax) and soft cloth is all a non-repair-shop-trained person EVER needs to clean a brass instrument.
Nearly all brass valves are PLATED, not solid metal. If you wear down the plating, it will either no longer fit in the valve casing properly or it will no longer fully seat correctly - meaning it'll get stuck more often.
If you use it on the piping, you can wear down or deform the metal with scratches or microabrasions. You're doing more harm than good.
To OP: If a mouthpiece is stuck THAT badly, it either fell off of the stands or was somehow kicked. They SHOULD come out easily - unless something is misshapen. Your repair shop is going to use heat and penetrating oil to remove it, not pressure that may deform the lead pipe. You should absolutely do the same.
If you don't have a heat gun (everyone should, they're about $20-30 at Home Depot, or you could buy the Valentino kit that comes with one) then even a hairdryer on "hot" aimed at the leadpipe and gentle taps with a rawhide mallet *at the point of entry* should be enough to remove even the most stuck mouthpiece.
If it's not - if it's in 1/8th of an inch or more "too far" then you've likely deformed the lead pipe and need to get it repaired or replaced ANYWAY, so send it to the shop.