Ok, I must have been thinking of syncromesh which no longer required rev matching and double clutching. I had it in my head that syncros allowed the gears to match engine speed, but I see I was wrong.
That is what syncros do, they syncronize the speed of the input and output shafts allowing you to smoothly change gears.
The issue that seems apparent here though and what most people have an issue with internally to VW transmissions is not actually the syncros themselves. In a way it is the syncros but not really. What you probably are experience is damage to the selector hub which is why it is notchy and grindy when not even moving. If you pulled the syncro out it is probably shiny, new and even reusable (for the price you just always replace it).
These parts are actually out of a transfer case but they make the point and are similar inside almost any modern syncromesh transmission. I happened to be building a beefier TC for my cummins wrangler last week so had fresh pics on my phone.
The part to the upper left would be the same as a gear in our transmissions. There are two parts to a gear there is the gear that is free spinning and a gear that is fixed, this is the former. When you are not in gear they are still in constant contact however the spinning gear is not locked to the shaft and no power can be transmitted through it.
Then below that you have what would be the same as a selector hub in these transmissions. When you move your shifter inside it moves a cable which moves a rod on the top of the transmission which then operates each of these selector hubs. They are almost always going to be paired in what would a front to back movement of the shifter. So 1-2, 3-4, 5-6.
The selector hub is fixed to the shaft as you can see by the splines in the center of it. There are usually spring loaded tabs that go into the rectangular notches you can see in 3 places on the hub. In this case the older small one didn't have them the new bigger one did. In this sort of application (the transfer case) it is not 100% necessary for a standard transmission it is very important or you could end up with a car in 3rd gear and the hub for 1-2 partially engaging either of those gears. Even if it didn't engage those gears it would most likely grind and cause wear.
On the bottom right piece you see a gold brass ring, that is the syncro. Again looks like that in almost every application. Sometimes there will be a steel ring and then another syncro which VW often does on second gear. Often times a shift into second is the most hasty so they are basically trying to even bring the syncro into speed with the input and output shafts as quickly as possible, the additional brass ring is basically a syncro for the syncro.
Now looking at all those parts notice how they all have on some part of them a similar toothed pattern. Not teeth like on a belt or a gear but either a triangle shape sticking out or a triangle shape cut into it. The latter is hard to see but if you look at the lower right where the brass part meshes with the steel you might notice it. That is really all the contact area there is between the hub, syncro and free spinning gear when that gear is engaged and locked with the speed of the shaft.
Also hard to see in this pic is the shape of those little triangles, that syncro is not flat on the opposite side it has a peak like the roof on a house.
ok google to the rescue for the visual:
oddly yet another transfer case syncro but this was the best pic...
That little roof peak shape is actually what is doing the syncronizing. The selector hub has little peaks just like that lined up so the point of the peak is to the point of the peak on the syncro. However you don't want it to hit peak to peak that as just an explanation of how they work. What you want is those angled sides to slide together. This brings the hub into speed with the syncro and since the friction to the free spinning gear is higher with the syncro then the hub to the syncro it should be the same speed following closely behind the syncro engaging with the hub. The free spinning gear would have little peaked roof teeth also facing the same direction as the syncro.
So what was the point of all this..
If you have regular or constant grinding in just one gear and no issue with the gear that shares the selector hub then chances are you have a problem with the syncro "system" in terms of how it is supposed to function but i would say 9 out of 10 transmissions i have opened that syncro has looked brand new. The hub and the gear though are beat to snot their peaks rounded down to a flat roof no longer able to engage properly. An issue with the syncro not really doing its job but not really a failed syncro.
When you hear grinding its those little peaks just sliding across each other never actually slipping down to either side and locking into place. It isn't the actual gears like most people think because they are in constant contact anyway, the noise is your hard steel gear against the hard steel selector hub.
What the OP is describing since he is having an issue even when the vehicle is stationary seems to be with the hub not the syncros. Missing a gear or even just shifting hard can put a lot of stress on the sliding part of the selector hub. It is a very precision ground piece that is floating on those spring tabs I mentioned and fluid. When you miss a gear though and those roof peaks smash each other that sliding portion cocks and digs into the fixed center part of the hub. Over time this deforms the metal and can jam it up. This will make any gear selection done with that hub difficult and make it apparent at any speed sometimes even engine shut off.
For a repair understanding how all this works can make sense so you know what is all being replaced. If a shop can't show you what is failed take pics and get another opinion. If they are just throwing parts at it find out why. If they are just replacing the syncro even though that can save you hundreds of dollars that day it may not actually fix the problem and the trans will be back out and in pieces within a week costing you double in labor.
OP since you know this happened going into second if this was replaced in the near future you could probably get a way with a 1-2 hub, second gear syncros and maybe not even second gear unless you have been having grinding issues. Let it go too long and it may take out first gear itself but any metal floating around can affect bearings, other syncros (slowing them down so you damage other hubs/gears), and even cause wear on the gear faces themselves causing noises down the road.