How so? People circle eachother and maneuver to try to get an advantage in a melee.
Yes, cartoonish you often end up with a line enemy-PC-enemy-PC-Enemy-PC then the next enemy to join move behind the PC on the back to get advantage. Eventually you hit a wall or shrub or something and that is what stops the practice.
No, it doesn’t. With Blur and invisibility it’s still worthwhile to turn advantage or disadvantage to a flat roll using flanking. Barbarians get to get advantage while tanking something solo, which others can’t usually do. That’s still the case if others can flank for advantage. Dodge is an infrequent choice for a reason, but it shouldn’t be any less frequently used. IME, Monks use Patient Defense more when flanking is possible and they’re outnumbered.
Mathematically It is a huge nerf. Disadvantage with a high AC will reduce the chance to hit by up to 95% over a straight roll. Blur or invisibility with a target 20 to hit takes the chance to hit from 5% to 0.25% or one hit in every 400 rolls. Give out advantage to flatten it and it is back to 5%. You will literally get hit 20 times as often. That is a far bigger impact than going from normal to advantage in the same situation where you will only get hit about twice as often.
To put this math in perspective if you face 100 enemy attacks over the coarse of a day that do 1d8dmg each while you are invisible with a 20 to hit your AC, on average you will take 2 points of damage if the enemy has disadvantage and 45 points of damage on average if he doesn't. That is a HUGE nerf.
WRT the Barbarian, you are taking the key feature and giving a lot of it to everyone. Yes, a Barbarian can still use it solo, but much of the time everyone can easily do something only he could do. This is even worse when you consider Barbarian is an underpowered class compared to the Paladin, Fighter and Tasha's Ranger.
Why would a monk use patient defense instead of simply attacking with advantage or taking disengage as a bonus action and moving out of range (since their move is higher than everyone else's).
Oof. That’s a mix of DMs not playing the monsters tactically to encourage tactical PC behavior, and just player mindset. It certainly doesn’t apply to my group, or other groups I know that use the rule.
My big thing is it does not improve the game at all. I have played a lot of games both ways and IME it brings nothing at all to the table IME. You want advantage use the help action or shove prone or come up with something creative. They can still get advantage, even at will, it just takes away and attack or an action to do it. That is what smart monsters will do in the example with invisibility that I used above, instead of just moving to the end of the congo line.
YMMV, but I think in general more people who have used it have had bad experiences than have had good.