Can someone with a half Thai child living abroad get a visa to stay in Thailand?

nope, 😕 the "reason" for your visa/extension has to be here when you apply. Same for a marriage visa/extension at the immigration office you need your thai wife with you when you apply.

You MIGHT <- (as in may be able to) apply for a year-long multi-entry Non-O visa based on raising a thai child at a thai embassy before you come here...

No consulate in S/E Asia issues that year-multi-Non-O except the one in Penang Malaysia and you need to bank 200K baht in an account along with the required paperwork to get that one issued