Flamenco in Madrid, Seville or Granada?

Stick to shows in Seville or Granada.
Stick to places that do not serve food.
You can use this site to see offerings in each city and to book tickets:
https://flamencotickets.com/seville-flamenco-shows

I recommend to book no sooner than the day before you decide you want to see a show.

Flamenco everywhere!
Each dancer (/singer, musician) is unique, and offers a different artistic interpretation. I prefer not to recommend any specific place anymore, because what people (tourists) enjoy is subjective, and I love ALL flamenco (I also love to dance it a lot!). There are many reviews and lists of flamenco shows on this website and other websites.

I disagree with avoiding places that serve food--I actually find it quite enjoyable to have my tapas and my vino on my little table while I spend a few hours sitting and watching great performances.

I saw an excellent show two weeks ago in Sevilla at the MUSEO DEL BAILE FLAMENCO. Small venue so view lines are excellent.

I love flamenco! Every visit to Spain, I go to a show. I've never been disappointed. I haven't found that the location makes much of a difference. All the performers give 100%. Base your decision on how best an evening show will fit into your schedule.

As for the touristy aspect of the audience, I find this to be inescapable. I've never attended a show without at least one tour group being present. Just focus on the performance, and you won't notice the people around you.

Best seen in the south that is where Flamenco originated.

I followed a Rick Steves recommendation to see authentic flamenco in Granada in 2005. It was in one of the cave museums across the valley from the Alhambra. We arrived early and got Castnet practice and lessons and also danced flamingo with the dancers. I would recommend this experience over a staged program any day.

Granada, with option for dinner beforehand -- https://flamencogranada.com/en/

Sevilla, with museum -- https://museodelbaileflamenco.com/index.php/espectaculo-patio-flamenco/

Sevilla, free performances -- Plaza de España

Don’t count on there being free performances at the Plaza de España in Sevilla. Most likely you’ll find the Roma selling fans and trinkets.

In Sevilla, we went to Plaza de España twice and saw flamenco performances each time. They passed the hat around, and we gave them 5 euros.
Here is a website describing these performances -- https://thenextcrossing.com/plaza-de-espana-and-park-maria-luisa

We also went to a flamenco show at Museo del Baile Flamenco in Sevilla, which was a great show. The museum there is also very good -- https://museodelbaileflamenco.com/

In Granada, we had a very good dinner at Jardines de Zoraya, and watched their excellent flamenco show afterwards. It is high up the Albaicin hill, so we took a taxi up there. Walking back down later was fine. Their website -- https://flamencogranada.com/

We saw a flamenco show at Cafe Ziryab in Madrid, it was awesome and we had 2nd row seats! Unfortunately it was our only flamenco experience in Spain, but it was such a wonderful and intimate experience that we'll catch flamenco anytime we go to Spain.