So, the fact that laser printers work so well is that you can directly roll-on the toner – that either requires a very finely tuned linear transport of the paper to be printed on (which is mechanically complex and thus expensive), or, and that's what you find in any paper printer – keeping the paper under the right tension and having another roller opposite to the toner transfering roller to which you feed the paper under an angle (so that you can keep it flat and tensioned). Another aspect of bending the paper around rollers is that it makes it possible to put relatively long processing chains within a compact case.
Same goes for inkjets – you need to make sure your piece of paper is 100% flat where you run the ink nozzles. That, together with the need to grab a single sheet of paper, make linear transport expensive to build.
Of course, there's speciality printers, eg. for surface printing on metal or plastic signs, or of course, cardboard postcards, packaging etc. But these are large (thus making the bending radii large) or expensive, so that even there, the "stick-on"/label approach is often preferred.
If you really can't stand the labels (which don't sound too bad, to me), I'd look into getting a cheap laser cutter/engraver and running it with as little power you need to burn in the info you need :) You'll need to write a few scripts to put the text where you want it, convert it to something your cutter speaks, send that job out to the cutter etc, but bonus is you get a laser cutter for other purposes :)