What about Valencia as a destination?

The distance from Madrid to Sevilla vs. Valencia is negligible, with the AVE high speed train - 2 hrs to Valencia and 3 hrs to Sevilla. I would go with Sevilla, if this is your first time around these parts.

Been to both, both make very nice visits. I would say that Seville has more beautiful sites and neighborhoods, also flamenco. In general, I would choose Seville, unless you are going during the summer when that region can be very, very hot. Córdoba is between Seville and Madrid, and is the site of the wonderful Mesquita. Córdoba is a terrific visit, for a day, or better for a night or two. All good choices, if you can avoid 100+ degree heat.

Logistically Valencia makes sense but Seville beats Valencia hans down in my book. But if you aren't including Cordoba and Granada, then skip Seville this trip for an Andalucia based trip next time.

If you want to minimize long distance travel, what about adding from Madrid
- a night in Toledo (or a day trip, but it's less good)
- a day trip to Segovia
And from Barcelona
- a day trip to Gerona (or a night)
- a day trip to Montserrat

All of these destinations are more interesting than Valencia, which is a city I very much enjoyed when I spent a long weekend there 3 years ago, but would not want to cross an ocean for (I assume you live overseas).
Of course, if the places above are already in your schedule, then why not Valencia.

My vote is for Sevilla. Have been to Valencia twice and found it uninspiring.