Would you visit Madrid in early October?

Yes, early October should be quite nice. Average high is 21C /70F for the month, and the early part of the month will on average be warmer than the end. I've been in mid-October and it was around 60-70F, people were out and about, sidewalk cafes were popular, and it was basically perfect weather for city walking. Too cold for a beach or pool, but Madrid isn't the city for that anyway.
Obviously weather is variable and you may hit an abnormally cold year, but the averages are in your favor.

My last visit to Madrid (2009) was at this very time (mid October). I carried a light sweater with me every day, and ended up just doing that, carrying it. Lots of sun. No rain that I recall ... or that my photos reveal.

I was there in September 2015, and wore short sleeves every day; it was nice weather, but a bit on the hot side (by day) for long walks. So, October should be great weather, not too hot or too cold to enjoy.

When I was there in early October of 2011, I was wishing for a pool and probably never wore my sweater.

I'd expect warm weather - shorts, sandals, sunhat - and be pleasantly surprised if it dropped a high of only 20 for a day or two.

Consider 1-2 nights in Toledo for a change.

I was there last October; twice on one trip; at the beginning and end of the trip.
At the beginning, around Sept. 30th, it was hot and sticky.
10 days later, I was wearing every piece of clothing I had with me to keep warm!
It was sunny at the beginning, then at the end was rain and wind.
So take layers, and a very thin pair of long underwear that can go under long pants or even act like leggings under a skirt. I was even cold in my hotel room!
The museums are great, and I was lucky to be there on a Free Entrance day, so got to see two of the big three for free.
I really liked the Thyssen-Bornemisza Gallery the best.
Layer up, and go!
Great coffee in Spain!