How prevalent is the dock scratching the Nintendo Switch's screen?

Don't worry too much about this. I'm pretty sure a big part of the "scratch fiasco" is websites and YouTube channels looking for ad impressions (and money), unfortunately. I've removed/docked my Switch several times since day 1 and I don't see any scratches or even smears/traces on my screen (with a screen protector).

Looking at my dock, I simply see no way the dock alone should be able to scratch the screen, because there are two plastic guides with small rubber pads to keep the screen away from the dock, but they don't really have any sharp edges.

Of course they might in theory still scratch the screen, especially if you're just "throwing" the console into the dock or if there are bigger dust particles in there (maybe even sand from your hands?).

As such I think adding any good quality screen protector should offer you enough protection. It doesn't have to be tempered glass, even just a not too thin foil should be sufficient. If you notice damage on it, replace it, otherwise just enjoy playing.

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Edit: Also, something I noticed the very first time I saw that now "famous" picture of the scratches: Their orientation is completely inconsistent and doesn't make any sense, if you assume the console is inserted in a straight way as intended. Just look at the scratches highlighted with a few lines:

This looks – at least to me – very deliberate, rather than random usage. Also wouldn't you assume there to be one or two scratches before you notice them? Rather than waiting for 10-20 to appear?