For a broad overview, I guess you're OK. The thing about Barcelona is the (even pre-COVID) need to have timed entry tickets for so many of the top sights. That has made it a difficult to city to visit at a dead run, because you risk either cutting your time too short at those (often expensive) sights or building in too much of a buffer between them. I'm not sure you'll have a lot of extra walking-around time. But that does depend on which places you think need an inside visit. It works best to slot one of the must-book-ahead sights first thing each morning, but you'll have only two mornings to work with.
The good thing about counting on a future visit to see more of Madrid and Barcelona is that in normal times you can get non-stop flights to them from quite a few US origins, and they are often reasonably priced. Flying in or out of Seville tends to ratchet up the airfare a lot, though it's well-connected to Madrid (and Cordoba) by train. Granada at this point is not quite as easy because there aren't as many fast trains per day, which limits your flexibility a bit, and flights into/out of Granada tend to be pricey.
I hate to see you try to visit Granada on a day-trip from Seville. That's a lot of hours sitting on trains and not much time to enjoy Granada. The Alhambra is not a 2-hour sight unless all you want to see is the Nazrid Palaces. There's a lot more to the complex than that (though they are the #1 attraction), and the city itself has other sights as well as atmospheric areas to wander. I yield to no one in the degree of my hatred of changing hotels, but you can make a nice loop from Madrid/Cordoba to Seville to Granada. Definitely at least look at the possibility of starting your trip back home from Granada rather than from Seville; perhaps for your home airport there won't be much difference in the fare.
It is likely to be very, very hot in Seville in June and unpleasantly hot in Madrid and Granada. Being mentally prepared for that will help. Statistically, spring break would be a nicer time to visit the south, weather-wise, but Semana Santa is a very popular time to visit Andalucia, so you'd be facing considerably higher lodging costs at that time of year.
It's also worth pricing out flying into Granada or Seville (changing planes somewhere--ideally in Spain--on the same ticket). The earlier in June you hit the southern cities, the better, unless you uncover some sort of special event you want to align with.
Can you not add a few days to this trip? It would help a lot.