You don't seem to be getting my point.
The TSAB can't get the JS out of Shirou. Because of the interference by this mysterious artifact (Avalon), they can't properly seal the JS, and because they can't properly seal it, the interference persists. Every time the Jewel Seed has acted up, it has been because of Reality Marble shenanigans that make it act up and that Shirou only survives (probably) because he has Avalon.
If Avalon tries to come out, then the Jewel Seed will react just the same as when people try to take out the JS. Sealing Avalon for long enough isn't impossible, probably. Doing it with Mid Childan magic? In any reasonable amount of time? Not so much. Assuming they succeed in taking out Avalon? They still have to deal with UBW. And that RM is another OCP problem they can't really deal with.
The driving point during the first season of team Yuuno and whoever was on his side was Sealing the Jewel Seeds before they spontaneously activated and caused a major disaster (think dimensional quake, if a JS chain activation happened, or if one JS received sufficient stimuli). The Jewel Seed in Shirou won't act up unless Shirou's RM stays out of the game, but it has woven itself so tightly to Shirou that any attempt to remove it will cause unworkable to enter the game.
The TSAB being good enough to safely seal the Jewel Seeds in certain situation don't mean that they should be able to do that in other situations, even if you think they should, be it for the sake of the plot or for internal consistency. With the TSAB, it's not "Jewel Seeds are safe because we can seal them" it is "to make them safe, we must seal Jewel Seeds while we can, and before they cause disasters".