Well, I guess that theoretically you could, and it would likely be good for taming, but in my opinion, it really isn't a good way to take the bird out on a permanent, long-term basis.
Having a bird takes responsibilities to him/her like making sure the bird has enough room to spread its wings, which includes letting him/her our every day for a few hours in a bird-proof room.
Small birds (tiels, budgies, etc.) don't make too much of a mess when they're out. If they do poop, it's easy to clean up. The only time birds would really "make a mess" is when they moult, in which case you'd need to vacuum the room regularly to get rid of dust, dander, and feathers.
If your mom is concerned, explain to her that you'd be responsible and clean up after him/her and stay true to your word.