I did a bit of research, and it turns out this is not the case in Japan, where I bumped into this issue. Japan Merchant Services and Mitsui Sumitomo say they will not accept handwritten slips prepared from unembossed cards for processing.
If this was some small country that barely saw any tourists I would have just left it at that, but I figured it bears mentioning because Japan is a fairly popular travel destination, and the 2020 Olympics next year will mean even more visitors going. So yeah, bring an embossed card (good thing Monzo is).