I do the same with gloves although mine are medical grade vinyl type. I do so for two reasons. One is protection of my skin for I do have a sensitive skin. Second is to avoid my contaminated hand from staining/contaminating new paper in box/safe. Although I wash my hands before handling unexposed paper, I'm afraid trace amount or drop on my hand may contaminate it. Now, I just remove my gloves, quick rinse and dry, and I'm ready for the second print.
As to OP's question....
For anything 11x14 and larger and FB, I have problem with getting all surface of paper contacting developer at the same time. This is especially true with FB because they curl toward the emulsion side. I always have some part that refuses to get "wet". To avoid this, I put the paper emulsion side down, slosh, slosh, GENTLY push with my palm, until the paper become saturated then flip it over. Then keep rocking the tray slowly and gently. I dev 2 minutes from the point I flip it over. Usually 2.5 to 3 minutes total. Longer if dealing with warm tone paper. It is REALLY easy for paper to float up and not get enough developer, so I keep watchful eye over it during the whole time.
To dry, I spread bath towel and put my washed and squeegeed prints face UP. Works for me.
As for agitation, although I only print up to 12x16 I put a thin piece of wood under the dish and use this as a fulcrum to rock the dish gently back and forward. The undeveloped print goes in face up and on the 1st 'rock' the developer flows evenly all over the print. No air bubbles have a chance to form. For even better coverage you can always add a small amount of wetting agent (Ilford Ilfosol or Kodak Photoflow).
I do agree with giving the print more time than 2 minutes, we pay for the silver content, so let us use all of it. I also use another technique where a highlight is a bit stubborn, and that is to 'paint' the light area with undiluted developer using a 1/2" brush. The same applies to the shadow areas too if you want a touch more depth. It isn't a big difference but a one that can rescue a print from the scrap bin
I too have recently printed my first 16x20 prints. I really dislike printing this large. Even though they are printed from 4x5 negatives and my entire workflow is dedicated to producing the sharpest negatives possible, they are just a little softer than I like to see. I also think the inverse square law applies to larger prints: the chances of everything going just right are inversely proportional to the square of the print size. 11x14 is my favorite size to print.
I submerge the print by sliding it quickly into the developer and then agitating using rubber-tipped print tongs (I have bamboo tongs that have been going for 20+ years!). A side or end of the print floats up in the developer and I gently press it down with the tongs. Moving from area to area doing this throughout the entire developing time provides constant and even agitation. Often, I'll flip the print face down for a bit and then face up again. I'll use two tongs for larger prints, but manage 16x20 usually with only one. The tongs allow me to handle the prints all the way through the fixer without contaminating my hands. Never damage to the paper surface this way either.
When developing more than one print at a time, I use my hands and shuffle the prints. I do not use gloves for common HQ and PQ developers, but do for amidol and glycin developers (and for pyro negative developers). When shuffling, I often get some emulsion flaking on the very edges of the print where I handle the paper. For me this is not an issue, since I print with generous borders and trim them off when mounting. When I shuffle prints, I am careful to wash and thoroughly dry my hands before returning to the dry side for more enlarging.
I find tray rocking inadequate for agitation personally, but there are those who have mastered the technique of setting up enough waves in the solution to do the job. I've simply never found it necessary. I have not had problems with print damage from tongs or fingers ever.
As for developing time: Your 1'40" is much to short for optimum development of most fiber-base papers. That, coupled with your possibly inadequate agitation is likely the cause of your density variations (although, it could also be enlarger-light fluctuations, especially if you are using a cold-light head).
I use 2.5 to 3 minutes as a starting point for fiber-base papers and often develop for up to 5 minutes. Try the three-minute developing time recommended above, increase your agitation (more can't hurt!) and see if that eliminates your variation problem.
As I recall the hoity-toity art photography schools, whose students worked with large prints a lot, never believed in flipping the print over (air bubbles). Have you tried simply grabbing a corner with tongs and hand-agitating the print? To this country boy your agaiation triple-header sounds like a dog-and-pony show. Do you tap dance while all this is going on? I think you need to streamline and smooth out your agitation. YOu're giving your prints cardiac arrest.
I never wear gloves while doing regular prints and my hands keep dry during the process. Taking the paper from one tray to the next is easy with a tong for smaller formats. But for bigger paper sizes a tong may not be that practical. Therefore and due the small size of my darkroom, when printing that big I do one tray development. So I place the paper into the dry tray and then pour the developer over the paper - that way I also can make sure that all the paper gets covered with developer within seconds. For FB I develop three minutes with constant agitation, then I pour back the developer into the bottle (using a funnel). Then I pour in the stop bath etc. Since I rarely print that big, I do also all the washing, toning and again the washing in that one tray, so the first time my fingers get in touch with any liquid is after the final wash, when I take the paper out of the tray.
That might sound impractical for you, but it works very well for me.
I always slide the print in face up, get it saturated with developer then turn it over and slide it in face down. No problems with unevenness. I develop FB prints for at least 3 minutes, sometimes 4 or even 5. I have had stains on both MGIV FB WT and Adox MCC110 from fingerprints, though the MGIV is much more susceptible. I use tongs but they do get awkward with 16x20 and sometimes require a helping hand, and this has (but far from always) resulted in stains in fingerprint patterns. I find gloves almost intolerable. I'm not worried about exposure to occasional small amounts of the usual developers (I would always wear gloves if I used amidol though.)
No matter how you do it, 16x20 is starting to get awkward and easily damaged for standard double weight FB paper. RC is a bit easier in this regard.