Tom - So.... starting with a lock that has already been assembled, you may find that you kind of have to pick your battles. I don't want to suggest that commercial locks are generally inherently flawed and always need to be worked on so they function properly, but sometimes they can be improved.
So what do I mean by picking your battles? Well, sometimes you'll hear people say that the first thing you want to do when reworking a lock is to straighten and flatten the plate. What if it's bent and / or twisted to the degree that straightening and flattening is going to change relationships between say the group of internal parts and the mainspring, or between the group of internals themselves? If the lock is working well, i.e. it was assembled that way, maybe it's best left bent and unflattened unless you're prepared to do a bunch of other work.
Ideally, the plate would be flat and true, all internal parts would be mounted square to it, and your bearing surfaces would have a nice slip fit - but in truth, flintlocks can still work extremely well if there is a little slop here and there. How much slop is not an easy thing to quantify. Generally, the closer you get to the ideal the better since the lock will tend to wear more evenly and perform well for a longer time.
With that in mind, either flatten the plate or decide to leave it alone. From there, look at how your tumbler shanks fit in the corresponding holes in the plate and bridle. Much slop there will make for erratic performance. Ideally you want the tumbler shanks / axles cleaned up by turning and polishing, and you'd then drill and ream the holes for a slip fit. Sometimes locks are assembled with the tumbler axles basically as cast. If the axles are rough, the first thing to do is to clean them up. At the same time you can create smooth, true bearing surfaces around the shank where the body of the tumble r will contact the bridle and plate. This bearing surface only needs to be a small ring around the shank, not the entire side face of the tumbler.
You also usually want to clean up and polish the inside of that bridle since it is a bearing surface for both your tumbler and sear - but since you have other issues to address this may change. More on this in a minute.
From there , if you have an much slop between axles and their holes (again - hard thing to quantify but the ideal is a slip fit) you will want to install bushings to take up the slack. I usually make them from bearing bronze. I don't have a milling machine, so to install them I use a drill press. I also don't have any fixtures for holding the plate, I just set it on a block or frame and shim it square to a chucked reamer in 2 directions then epoxy or bondo the plate temporarily to the block. Looks stupid, works great. The block can be clamped in position and parts removed and added / drilling and reaming done so everything stays lined up.
You say there is a bunch of side play in your tumbler. Does your sear also have a similar fit? If so, then yes I'd carefully reduce the height of that boss you showed in the picture. I do this by hand filing. As you bring it down, you'll see what needs adjusted and where. In the end you want all contact surfaces polished and operating smoothly.
If however your sear fits well but the tumbler has side play, you may do better by intalling a tumbler bushing in the bridle that protrudes slightly inboard. You then fit that down until everything is tight.
Not to get too far ahead, but also take a look at the fit of your sear screw . the ideal is to have a slip fit between the hole in the sear and the screw shank, and you want the threads cut to the proper length. If the threads are cut too long up the shank, when tightened the screw will flex the bridle inward and bind the sear. Ideally you time the threads so the uncut shank stops on the plate just as the screw head snugs against the bridle. Similar arrangement with the frizzen screw.
