Maybe I can make a few suggestions.
I suspect all that noise you mention is being generated from multiple sources. Fresh spindle and motor bearings should help to reduce some of it. Even better in my opinion would be to go with a 3 ph motor and VFD combination. Yes a fair amount of cost, but the lower end single ph motors used on these lathes are built down to a price point as well. And 3 ph motors simply operate much smoother and quieter. It's mostly why there universally used on any industrial level machine tools with even fairly low HP motors. I'd also bet that lathe has slowly gotten a bit noisier over time just from added wear on items such as change gear bushings or those stub axles they run on. That causes unavoidable and variable backlash on the spur gearing. Anything you can do to reduce that wear on the gear bores or those stub axles will help a lot. Again those assemblies are built down to meet a price point, and for most of these lathes, it definitely shows with that extra gear noise. Properly re-boring spur gears to add anything like oilite bushings or better, is still touchy work to do it right. In my opinion, a decent 10ths capable DTI to align and re-bore truly concentric to the gear teeth really isn't optional. Shimming or building custom machined and sized washers to fit against the gear faces will lessen the side to side clearance float and help with some of that extra noise.
Some of these lathes come with a couple of plastic gears, many seem to think those are inferior and buy steel replacements that are being made available for some of these lathes and mills. Engineering wise, there's some good reasons for the manufacturer to chose that gear material. It's in general a cheapest part first after the shear pin situation in the event of a serious crash, and they help the machine to run quieter. Myford for example could have easily produced there tumbler reverse gears in cast iron or steel for there Super 7 lathes, instead they used phenolic to help with that noise and as a secondary fail safe. Lots of other manufacturers do the same. I believe Emco used at least one or two fiber change gears on there fairly expensive gear head Super 11's. Even my little South Bend shaper uses phenolic on its bull gear to help quieten that noise.
If you can drastically reduce that gear slop back to acceptable levels, then setting a known gear clearance for backlash between each pair of gears can also help to quiet them a lot. For any that don't know it yet, I've seen some on Youtube just squeeze the gears together by hand and then tightening them down in that position. What that does is bottom the gear teeth against each other which they were never designed to be set or operated like that, and it creates high rates of excess wear as well as that extra noise. For spur type gearing, there's actually a cheap, quick and easy method to set that gear lash to a known and repeatable point. Simply use a fairly narrow strip of sheet paper between those gear teeth as there squeezed together and fastened in place. Since paper products can be found in a whole variety of thicknesses, experimenting a bit with different amounts of gear clearance can quiet them even more. If it were me, and if you don't already do so, I think I'd try a strip of normal printer paper first (about .003") and then work in either direction to see what your own particular set of gears works the quietest with. I sometimes use the very old machinist trick of cigarette rolling papers as edge finders, my Zig Zag brand measure for all intents and purposes at .001" thick, folding it or using more than one can get you paper thicknesses in .001" increments.
Fwiw and since I mentioned those gears. I've also seen many advocating the use of grease or even motorcycle chain lube products on there lathe change gears to both reduce the oil mess being thrown off and help quieten the gears a bit. Unfortunately either of those products will help to retain any contamination and wear particles created from just using the machine. And those are then being continuously cycled back through the gearing creating even higher rates of wear. In reality there's simple and unarguable physics about what lube products you should be using and where. You also don't need a flood of oil, those spur gears when compared to something like an automotive transmission are under pretty light loads, a few drops of oil each time the lathe is used are all that's needed.
I've never been able to re-find the South Bend WW II era advisory they published. It's somewhere on the net, but during that war, quite a few with home shops were producing short runs of parts for that war effort. The story I read was that inaccurate surface finishes and dimensions were becoming a real issue due to parts being rejected. Apparently enough so the Department of Defence contacted South Bend for some kind of easy and cheap solution. South Bend put there engineers to work on it. What they finally came up with was for the owners of those home shops to cast a heavy thick steel mesh reinforced concrete bench top to bolt those bench top lathes down to. The old drawings I saw gave the important dimensions for cast in stud locations etc for different models of there lathes. Why that helps is pretty simple, it adds rigidity and a large amount of mass back into the lathe bed. Now just how true that story may or may not be I can't say. But it's a proven idea. There's other methods as well. On my first little 5" swing lathe bolted directly to the bench top, part taper was easily measured. I finally added a 1" thick 95 lb piece of steel plate under it. In my opinion that totally transformed it. Far quieter, it was then down to a few 10ths taper over almost 12" of it's longitudinal travel (once it was properly aligned and leveled) And it could produce much better surface finishes with the exact same material and cutting tools.
The lathes surrounding sheet metal can also add or help to magnify that noise. Every consumer grade drill press I've had was especially bad for that. Something as simple as adding a small O ring just slightly larger than the screw diameter and between the parts being joined together helped deaden the rattles and extra noise. As I said, your issues are most likely a combination of problems and there's very likely no one single fix that will instantly cure all of it.
