There has been a movement (often called HIP - historically informed performances) which use old instruments or modern copies of old instruments for performances which are trying to create the sound qualities and techniques of the performances from the period of the compositions. So a Mozart piano concerto would use an early piano (much smaller and less powerful than today's concert grand), string instruments with gut strings, a lower bridge with less tension on the strings, different bow and bowing technique, valveless brass instruments, lower pitch, etc). Part of the issue is that early pianos didn't have the current full 88 key range, so they can't be used to play a big Rachmaninoff or Tchaikovsky concerto. Typically in these HIP recordings the orchestras are also much smaller.
I've never heard a recording of a piano concerto played on an instrument which predates the concerto. For example, the Brautigam Mendelssohn Piano Concerto referred to by the OP is (according to a review I read) a reproduction of an 1830 Pleyel fortepiano, which would have been similar to the instruments of Mendelssohn's time, not a fortepiano from Mozart's time (which would be smaller and less powerful and have fewer keys).
So rather than choosing an instrument because it was less powerful than a modern instrument, I think most pianists like Brautigam are playing instruments to emulate those of the time (along with orchestras with instruments of the time.)
One final point, there are a fair number of pianists who specialize in playing these earlier instruments, including Brautigam. They have to use a different playing technique than most pianists, and gain a reputation for their expertise in playing the earlier fortepiano. They are not competing with Danill Trifonov or Yuja Wang or Mitsuko Uchida or the myriad of great pianists who play the modern piano. I don't know of any pianists (there are some I would guess) who move back and forth between playing the modern piano and the fortepiano.
Larry