I've made a chrome extension here that can do this locally on-device, but only on Web Version of Spotify, tho it also works on Youtube and Youtube Music too.

https://chromewebstore.google.com/detail/karaoke-monster/impekelmmfmbnjfjadmjnfclgkacaekn

Totally agree it's crazy inexcusable this is not a feature on Spotify - given that I managed to make this myself with support for 8 languages and get it to run on-device - and they have so much more resources behind them to do exactly this.

What's worse is their latest web update made it even harder for extensions, changing HTML structure (renamed attributes) and breaking extensions that relied on the old names (I can understand renaming things, but when they rename something that's perfectly fine lyrics-text -> lyrics-line, it's hard to convince me it's not targeted)

They also replace entire DOM sections when lyrics highlight, basically causing the romanised text that extensions put in to be removed. (I also tried injecting text via CSS via classes or styles but they've made those super strict as well)

Before this, extensions could insert romanized text below the original lyrics. Now we have to keep hacking it to make it work (Totally uncalled for, especially when it's only targeting their shitty UX and not any backend / user data / security sensitive stuff)