In most cases, yes, they played their own stuff. We have letters, texts from magazines/newspapers... Different people tell us about them playing (and we start having recordings by the early 20th century)

Beethoven would have been capable of playing the #23. He really started to notice his deafness progressing quickly by the early 1800s (at this time he starts to change his withdrawing from social functions). He was coming to terms with it at the time he wrote that sonata (that might be why it's so... tempestuous). I guess he was still able to get an idea of what he was playing. He stopped performing around 1811. In the early 1800s, he was getting plenty of commissions (fancy people asking him to compose things for money), so he would not need to perform as much (and he was getting away from people in general). I don't know if he actually performed the #23 in public. There were several years between the time he finished it and its publication.

Beethoven started his career playing the piano. He frequently performed his works, and also would improvise on the spot.

Some examples of super famous composers not playing their own piano music: Tchaikovsky and Ravel. It became less and less common for composers to play during the late 19th century.

A recent example of a serious composer and a super serious pianist (a much more rare thing in our days than in Beethoven's) would be Frederic Rzewski. Check his variations on The People United Will Never Be Defeated, composed in 1975.