Vinyl “Pops & Clicks” eliminators

In my opinion, playing LPs under a dust cover induces more sonic degradation in the form of coloration than a tick or pop as forms of noise.  In my opinion based on personal experience, of course.  The sweep arm idea is an interesting one, however.  Do you have a specific brand to recommend? They used to be commonplace, but one doesn't see them marketed much these days.

Lewm7, I suppose if you had a particularly bad dust cover but in the case of my dust covers there may even be a degree of improvement with the covers down as they do diminish sound pressure levels within. Both my tables are also suspended which may make a difference. Many tables now come without covers or covers that are very difficult to use (not hinged). For something like a Clearaudio or VPI table you use 1/4" Lexan. You have a bottom plate which the table sits on and the cover is hinged to the bottom plate. There are several companies that will make these for you. Here is one  https://www.displaycasej.com/custom-audio-covers These guys did the one for my 30/12 and the workmanship is perfect. Now, even if there was some degradation in sound quality there is a major improvement is the lifespan and performance of your records which to my mind is more important. 
For the sweep arm this is the one I use  https://www.sleevecityusa.com/Antistatic-Record-Cleaning-Arm-p/tac-01.htm. It is a bit light weight but it works great. The bristles are conductive carbon which is critical. The Integrity arm is nicer but stupidly they use a camel's hair brush which does not conduct! Dumb to say the least. With the Sleeve City arm keep the back end high. You have to position it to the spindle just right to get it to track the whole record. I have a felt mat stuck by the table that I wipe the brush off with. Hook the wire up to chassis ground. I slide the weight all the way forward and crazy glue it in place. I put a little low strength loc tite on the counterbalance weight threads and use the weight to balance the arm. You want the bristles just touching the vinyl. They should not bend! To heavy and the bristles bend in the direction of rotation and just run over the dust instead of picking it up. I have three arms. Every once in a while I will clean off the brush with alcohol and switch to the other arm so the three arms rotate between the two tables. 
I know you have to trust me on this one Lewm but once you have a proper dust cover made you will never live without one again. You can throw your record cleaning machine away and with the brush you can throw your zerostat away. 

mijostyn

Contrary to popular mythology records come from the manufacturer perfectly clean.

That's just plain wrong. I've been in record pressing plants and seen the LPs waiting to be sleeved. These aren't dust-free rooms, so of course some dust accumulates. Some plants are much better than others, but it's easy to show that some new records are packaged with dust on the surface and - sometimes - fingerprints.

If you buy used records then a record cleaning device is mandatory. I do not because every one I have ever played was unacceptably damaged.

I'm sorry for your bad luck! My experience has been mixed, but I do have some LPs that I bought well used and they sound great. I think that's explained in part by my use of a fine-line stylus shape, which rides a part of the groove not likely seen by the original owner's stylus. And I do use US cleaning, which helps.

Once a dirty record is played it is done for. Thousands of PSI pressure literally melts the dust right into the vinyl surface ...

The notion that a stylus exerts thousands of pounds per square inch at the playing surface has been bandied about for years, but I've never seen any mathematics to support the claim, or to prove that a stylus can literally melt dirt into the record. Nor have I seen any physical evidence to support the claim. In fact, my experience suggests just the opposite.

lewm

In my opinion, playing LPs under a dust cover induces more sonic degradation in the form of coloration than a tick or pop as forms of noise.

I think that depends on the particular turntable and dustcover. I would have agreed with you back when I had an Oracle, but I don't find it to be true with my VPI.

To be fair, the VPI TNT dustcover rests on the turntable shelf, and isn't directly connected to the turntable itself.  So that may account for some of the difference.