You can see a lot of the shift mechanism by taking the clutch cover and clutch hub off. You may get lucky and see a problem there. My 630 is one of the best shifting bikes I have ever owned but getting into neutral takes finesse.
For the gears that work, yes, the shifting is almost imperceptible--just the lightest snick into the next gear. I'll take a look inside and see what I can fins.
yes,number 14. I dont remember if you need to remove clutch or not but i would definetly check that(its behind cluch cover). It could be shift star clutch mechanism too. If all screws are tight, then it could be safe to use your way. Not guaranteed... usually something more expensive will break if motor is used knowing that everything is not right. See what willie said before,check and see.
I didn't find anything that seemed off. I attached a small bungy to the shift lever to apply a small amount of down force, not enough to move the lever. With it in place, the bike shifted fine. I left it on for about 1500 miles and then removed it, and the bike continues to shift fine. I'm believing something was slightly out of tolerance and with a few miles from new it worked itself out.