@seamus452
DTA = Double Taxation Agreement (or DTT, Double Taxation Treaty) which is an agreement between 2 countries. USA has them with many countries, including Spain.
Typically, they ensure that you don't pay tax twice on the same income.
However, if one country charges a lower tax rate (I agree, calling this "generous" is a bit confusing!), you might still owe a bit extra to the other (that charges a higher tax rate).
Exactly right! Becoming a legal resident (by doing your EU Citizen Registration) allows you to live/work in Spain indefinitely, and access the Spanish healthcare and pension systems. But there's no obligation to do so. (And, as Irish citizens, it's possible there might be reasons to access the Irish systems instead.)
You don't automatically become a Spanish tax resident by acquiring legal residence. As long as you keep your stays below the trigger level, you would remain only a US taxpayer/tax resident. Lots of folks (including us) have holiday pads in Spain and spend most of their time elsewhere.
Making the appropriate contributions to the Spanish healthcare system entitles you to public treatment in Spain. But you can request an EHIC (European Health Insurance Card) which covers you when travelling elsewhere in the EU. If you want faster service, you can carry additional private healthcare insurance, or simply pay for private treatment as required.
I've ended up mostly in Bulgaria, rather than Spain. Therefore, I registered in the Bulgarian health system. My status is "unemployed" which is about 15 euros per month, and entitles me to free or nearly-free treatment. And they gave me my EHIC which covers me when I'm in Spain. An additional private policy in Bulgaria is about 500-1,000 euros per year. Spain is obviously more expensive, but it gives you an idea relative to the USA.
I'm not knowledgeable about USA health insurance, other than it's typically pretty expensive. However, I don't see any reason why you can't keep paying your premiums, if you want to maintain continuity. I suppose you should check the policy to see if living abroad is a notifiable "change in circumstances" and see what level (if any) cover is provided outside the USA.
My US health cover was massively expensive, even when I was significantly younger, and in perfect health. And even then it had sneaky exclusions and very high annual deductibles. So I dropped it at the first opportunity! :-) But if you have a great policy from your employer, then I can certainly understand the rationale for keeping it.