I'm also using MG Chemicals Ferric Chloride. I'm afraid of putting the used etchant back into the bottle. I worry that bits of the paint I used as resist will contaminate the solution. For now I'm just keeping the used etchant in a separate container.
FYI, I wouldn't worry about that, at all. I read somewhere that aside from contamination with solvents, there's not much you can do to screw up ferric chloride etchant. I would add soap to the list, if you use a bubbler. And w/e is in Kemtech 15% muriatic acid.
Flecks of paint, leaves, twigs, dust, don't really matter. I keep a lid on my tank, but it is outdoors. I get some occasional debris. I pick or strain it out when the moon is blue.
I suspect OP had issues with paint re-adhering to the board, as others have pointed out. But if you have tried clean copper and it's still slow, a couple thing I'm gonna do. Check the thickness of your copper. If you're using 2+ oz pour, there's nothing necessarily wrong. The etching will be slow and uneven.
If you're using 1oz or lower, and/or you feel like the ferric is just too slow, add some hydrochloric acid. The ferric needs a little bit of excess HCl. If it doesn't have any (and pure lab grade ferric won't have any), it will etch fine for the first tiny bit, but it will quickly slow down. If you are using only a little part of your etchant, and keeping the rest in the bottle, you are setting yourself up for that problem. Some peeps seem to prefer this method, seeing it as less wasteful. I've seen where some people are putting a little Ferric on a sponge and wiping the board till its done. Then tossing it. And thinking they found the best way to use it without "wasting" any. This is obviously retarded AFAIC. This is just a dude who happened to use a half oz board and thought he discovered magic.
If you don't want to waste it, just use the whole 1L* bottle of it. And pour it back in a bottle or next time. Trying to use a little of it at a time and then disposing of it is more wasteful, in the long run.
*Or w/e volume will reasonably fill your container without fear of spilling. You can try the whole, put the pcb in a bag and a few mL or ferric, but this is not creating less mess or using less ferric. This is just dumb.