@digital_nomad
Don't beat yourself up, emergencies happen. And I don't think it's a big deal, as you can easily re-do your residence applications.
To clarify: many get confused about NIEs (Número de Identidad de Extranjero) and TIEs (Tarjeta de Identidad de Extranjero). The former is your unique ID / tax number, and, as far as I know, it doesn't expire. The latter is the biometric ID card (which does expire).
If your residence permits expired several years ago (2018 or 2019), I think you just have to start again. Firstly, you should do the EU Citizen Registration step again (as a Belgian citizen) and get a new certificate valid for 5 years. Your EU registration step does NOT require a job; you need proof of address. proof of funds, proof of health insurance. You need to do this in person, in Spain, at an immigration office (appointment booking, cita previa is online). I don't believe these steps can be done at a Spanish Embassy abroad. The requirements are listed here:
https://extranjeros.inclusion.gob.es/es/InformacionInteres/InformacionProcedimientos/CiudadanosComunitarios/hoja101/index.html
Secondly, do the Family Reunification step again for your non-EU spouse. Requirements listed here:
https://extranjeros.inclusion.gob.es/es/InformacionInteres/InformacionProcedimientos/Ciudadanosnocomunitarios/hoja012/index.html
Of course, it's a bit of a pain, but I don't see that it will be particularly difficult.
After your first 5 years (around 2019), you would have qualified for permanent residence. But an absence of several years to 2023 would have been sufficient to lose that anyway. If you re-start this year, you can apply for permanent residence in 2028.
Or, in relation to your questions:
Is it possible to renew her NIE and regain her family reunification residence status? - No (IMHO)
Do we have to start the family reunification card process all over again upon return? - Yes (IMHO), but only after YOU return and do your EU Citizen Registration again.
Can we start this process while in the US (we are currently in North Carolina) - No. It's an in-person appointment at your local immigration office (Cita Previa).
I am not currently employed (my spouse is head of household) and I will need to re-establish myself back in the EU (i.e. look for work in Spain), but can show proof of funds (savings) and health insurance if required - EU citizen can relocate for any reason, job not required. You need proof of address + proof of funds + proof of health insurance.
I suppose there is a very slim possibility that the "family emergency" + Pandemic caused an unavoidable absence would be justification for allowing you to get permanent residence now. For this, you'd have to see an experienced immigration attorney and see if they feel it's worth trying. However, it seems very unlikely (and complicated/expensive) to me. In particular, while you didn't overstay in Spain, after expiry, which is good, you should note that there are residency requirements related to absence from Spain, and these were breached.
I think there's also a slight possibility that you could sneakily get back to Spain (drive over from France) and pretend you never left... and apply for a permanent residence permit on the basis that you've been there since 2014 (and EU passports don't usually get stamped so it might not be clear that you left). You need your little registration card. And the problem is that it expired 4 years ago. But I think they are more forgiving of EU citizens. It's a real long shot, but if you don't ask, you don't get. :-)