The person/organisation creating/requesting a DOI must definitely care because different object types need different metadata and each of the RAs has a specific focus on the kinds of objects it promotes DOIs for. Thus, users requiring a DOI need a clear understanding of who to request it from.
Distinctions are not relevant for resolution as long as there is at least one well-known resolver (e.g., at doi.org) that can resolve all DOIs no matter through which RA they were created.
This is the same as for Internet domain names - you need to know where to go to obtain/administer the correct one but after that it doesn’t matter.
Other services than simple resolution require you to know which RA created the DOI so you can choose the correct service e.g., for metadata search. Again, this is similar to the DNS system with its multiple whois services.
To some extent we already talk about DataCite DOIs, Crossref DOIs, EIDRs, etc. so what Ed proposes is already happening in practice.