I got a flat shortly after installing new Schwalbe high pressure rim tape. Turns out the tape immediately went a little bit skewy – is it bad enough to matter?

WTF ? This rim tape application looks pretty much perfect OP ?

To my eye from that photo I see a few things.
1. Rim tape is too narrow. I’ve had success with double wrapping tape that’s too narrow with the 1st circuit biased to one side and the 2nd circuit biased to the other side overlapping in the middle channel.
2. It doesn’t look to me like you pulled the tape taught while wrapping. It looks loose in places, but I’ve never used this particular brand.
3. It doesn’t look like you pressed down on the tape to stick it solidly to the rim and extract air bubbles. I only see a couple of what look to be thin strips of well stuck on tape down the middle channel of the rim. This step goes hand in hand with pulling it taught.
4. It doesn’t look like you tried this with sealant, so I’m unsure how you could know the tape doesn’t hold.
5. To be fair, I test my wheels overnight without sealant to see if my tape job holds and as long as there’s minimal air loss, I add sealant and ride. This assumes tires and any exposed rim aren’t leaky.
6. I double wrap my tape even when it’s the correct width for the rim, I’m paranoid like that.