The steering bearings occur in 2 places: directly beneath the top clamp inside the steering stem, and at the bottom of the stem inside, above the lower clamp. Originally they were ball bearings in races lubricated by thick grease. Over time, the grease dries out, and the bearings flatten or get "notchy" feeling.
Proper maintenance is to remove the upper clamp and the slide the stem out. Re-lubricate the bearings and re-install the stem. The stem needs to be tightened properly with some "lash".
To determine if your bearings are at fault, secure the bike on center stand, and grab the forks near the lowers. Pull forward and push back. There should be essentially no movement. Also, try racking the forks (pulling one, pushing the other). There should be no lateral movement either. (Technically there is movement, but by hand, you should not be able to initiate it).
If you've owned the bike for several years, then likely the steering bearings are in disrepair and should be greased. Many replace them with new tapered bearings instead as an "upgrade". To do this, remove the balls and races. The races will need to driven out with a punch or drift, top comes out "up" lower goes out "down". Takes a long drift to pass thru the steering head.
Not sure on your model, but the axle clamp (bottom of fork) should have an embossed "dot". That goes forward and actually creates a larger gap to the rear between the clamp and the fork. Snap a picture up close of your axles on both sides and post it up. Or, look underneath carefully and see if there is a "dot" and whether it's oriented forward.
"Gutentight" is probably "gutenuf" as long as you used the proper washers during installation.
Are the front forks stock length? If they are longer than stock, this will contribute to more oscillation and thereby a "wobbly" feel while less pressure and control is on the steering. You mentioned a recent tire replacement, that sounds like a great idea as tires over time become "flat spotted" in the bike is ridden normally straight. I always replace tires every 5 years, perhaps overkill. But I always have great steering and handling as a result.
My logic: tire is only piece of equipment in contact with road and thus is THE most important piece of safety gear.
Fork oil: there should be a drain bolt on the lower fork, probably on the outside of the leg. This will allow you to drain the oil. Bet it needs it badly. To replace, remove the fork top nut (carefully, there should be a spring inside that will need to be compressed during installation). Fill the fork with proper grade and volume (you can find this data in the manual). During draining, press the forks down to aide in expulsion of the fluid.
Some last things to check: the top nut on your steering clamps. Is it properly snugged and are the fork tube clamps (upper and lower) properly tightened? You don't provide much background on the ownership of the bike, so if you're behind on the maintenance intervals, all these items require attention.
If you're an auto mechanic, then you know all too well vehicles driven less frequently get overlooked more than daily drivers. We don't log mileage as closely and tend to put off maintenance under the guise of "I haven't driven it very much, it should be fine". On a vintage bike, this is quite dangerous. Any motorcycle requires strict attention to maintenance, more so than a car. Cars have redundant systems for safety, bikes do not. Spend the time going through the entire 3,000 mile service interval and be 100% all things are up to snuff. "Gutensnuff" that is.
