Madrid to Avila and/or Segovia

it appears a bus trip from Madrid to Avila is about 30 minutes

No, it is more like 1.5 hours.
And I would definitely prioritize Segovia over Ávila. A fast train goes there and makes it easier, but just one morning would be tight... it depends how late you are willing to take the train to Barcelona.

The Alcázar in Segovia is worth visiting as is the aqueduct. The wall in Ávila is nice and one can walk about 3/4 of the way around the town on it. There is a few Euro fee for walking the wall. In my opinion, the only other thing worth seeing in town is the church of Saint Teresa where her house used to stand. If I had to prioritize the two places, I’d choose Segovia then Ávila.

We had a rental car and from the airport to Ávila took about 1.5 hours.

Another vote for Segovia. There's just not much to see in Avila.

Possible, maybe. But reasonable…? To me, the sound of cramming in a day trip that day sounds exhausting and perhaps stressful.

A few notes:
-Note that different trains leave from different stations in Madrid. Trains to Segovia at Chamartin and Barcelona at Atocha. You probably wouldn’t want to have to cross the city after one train to get to another. (Not to mention the question of where luggage gets stored along that route… )
-Iryo trains will be quite a bit less expensive than AVE, but with fewer trains through the day.

Zaragoza could be a reasonable place to spend that day. It’s literally only the way between the two cities.

I can comment on Segovia....

In addition to Scudder's info, the fast trains go to Segovia Guiomar station, which is a couple of miles
outside the city and you have to take a bus/taxi to get to downtown. The local station only serves
local trains which take a lot longer to get there than the Renfe line.

There are frequent buses on Avanza that leave from Madrid Moncloa station and take about 1:20.
From Segovia bus station, it's walkable to the aqueduct, cathedral, and Alcazar.

So while I would say it is doable, there are multiple decision points and places where things can go awry.
Moncloa, for instance, is a fairly big station and finding your bus may require a bit of sniffing.

Going between Chamartin or Moncloa and Atocha via Metro will require a connection, which may be easy
or difficult depending on station connection and how much luggage you will have. I would see about storing
my bags near or in the Atocha station and then embarking on the day trip, whether it be by bus or train. Train
may be faster if the connections work out; the bus may be a bit easier as you have one less connection.

Everyone makes very valid points. I can see that I shall need to return in the future to visit Segovia properly. Thank you all very much for taking the time to comment.

On our first trip to Spain, we visited both Segovia and Avile. Skip Avila and enjoy Segovia!