I can confirm that in pretty much every instance where I've auditioned a turntable, playing with the dust cover down degraded the sound. At the extreme end it created feedback, at the least extreme it created unwanted vibration that, because the dust cover is typically attached to the plinth, is part of a path whereby dust cover vibrations transfer via the stylus to the playback system.
I never play an LP record with the dust cover down, and broadly speaking prefer a lift-off dust cover that does not touch the plinth at all, to be removed when the 'table is in use and replaced when it is not (blocking ... what else ... dust ... from the 'table's surface).
A well-designed cover attached to the turntable itself is not necessarily a problem as long as it is upright and stable during playback. I do prefer turntables whose hinges are such that they can be removed without marring the appearance from the front, so that if the owner wants, they could instead use a lift-off cover that is somewhat larger than the overall plinth footprint.
I think, James, that you have a very nice turntable offering there, and although $4K is a bit out of my budget (my turntable, cartridge, phono preamp and phono cabling I would estimate to be maybe $1500 worth, and isn't the weak point at this time) but definitely I see it as a high value, natural next step to what I do run these days.
The question is whether I will ever move to that next step, but that isn't the point; the product offering is the point and I have no problem saying out loud that I'm a fan. As usual, Bryston has managed to offer a High Performance product that is far from cheap yet is exactly as fine as it needs to be in order to offer excellent value.