The OED clearly does not associate the verb “welsh” or “welch” with any characteristic of people from Wales. It dates back much farther than 1905 – there are lots of examples from the mid-19th-century – and seems to have referred originally to bookmakers who absconded with the money wagered rather than paying off winners. That’s why the most common use is “welching on a bet” – failing to pay after losing a bet. It never occurred to me that this had anything to do with Wales or the Welch, although I suppose someone could have thought so at some point.
The OED is much more clear on the derivations of “jew” and “gyp” as verbs.