Depending on the material, color, lightning and size of the part - usually you don't even notice the layers, and in many cases even if you do it doesn't matter. For example I've printed big boxes for our bikes out of black PLA and they look fine - you wouldn't see it's 3d-printed unless you stare at it from 1 meter under good lightning.
And when you need a different finish you can always paint it. I've printed about a dozen plant pots of different sizes because my wife bought 10 ceramic pots and liked them very much and couldn't find similar ones anywhere. We painted them and they look identical and you don't see the layers under the paint.
I don't even print with the smallest layer size possible because it takes too much time and difference between 0.1 mm and 0.2 mm or even 0.3 mm is negligible in practice, but the printing time scales almost linearly.
Depending on what you use it for YMMV.
I've only printed a few things so far. Some board game tiles are perfectly smooth on the "top" (they were printed face down), and the very slight texture on the bottom makes a nice pattern when it reflects the light. The bottom texture is as coarse as a typical cotton fabric, but hard not soft. In other words, I can feel the texture, but it's even, like rubbing a pebble.
I also printed some chess pieces, which are essentially cylinders. The texture is visible, but completely consistent all the way up each piece.
If this battery door really does show the "limitation of an entry level 3D printer" then I'm very glad I spent a bit more money. I still know very little on this subject, but while choosing what to buy I found tutorials on building a printer from parts for €100 or so. This reminds me of the first print from something like that, before the expert showed how to correct the worst problems.
I bought one a year or so ago and while definitely still in the "toy" category, you can do some neat things with them. That said, woodworking is going to let you build way nicer things, but you don't get to just click a button and have it reproduced, nor share plans with others. But ya, FDM doesn't print things that are "nice" really, just functional.
To get good results, upload the files to a service like Shapeways and use their hugely-expensive SLS printers. You'll get your printed stuff in several days and the quality will be great.
The problem here is that you're zooming in on a 1x2 CM part...It's like taking a 12 Mpixel photo, then looking at a 320x200 size part of it...it'll still look pixelated.
Acetone vapor chambers are where it's at, closest you'll get to a injection moulded piece without doing the injection moulding.