Even without the piano (or even keyboard instruments), there were other forces pushing toward something close to equal temperament.
The common narrative is that chromatic music was instrumental in the use of temperaments that were more equal. While that is a factor, it's important to look at how chromatic music could be even in the 1600s. Frescobaldi's Fiori musicali, for one example, was an incredibly popular set of keyboard works published in 1635. However, when the more chromatic ricerars are performed on organs in the meantone temperaments popular in that day, they sound positively awful to modern ears that are not used to such wild deviations from equal temperament.
And yet, composers wrote this music and performed it on instruments at that time. Composers (and apparently listeners) appreciated the greater contrast provided by intervals that were more or less "in tune." I still remember the first time I noticed the compositional use of tuning for effect, where I was looking at an aria written around 1700 that briefly modulated through A-flat minor while the text was declaring something was "horrible, horrible!" Yes, the harpsichord accompanying that aria would likely have sounded quite "horrible" at that point in the piece with its harsh intervals that would seem very "out of tune" to modern ears.
So, strictly speaking, it wasn't necessarily chromaticism alone that drove the move toward equal temperament. Composers had been using chromaticism and tuning for effect for centuries before keyboard instruments came to be tuned in something approaching modern equal temperament.
The main thing I'd add to other answers here so far is the role of other instruments. Fretted instruments have already been mentioned as a very early use of equal temperament (long before keyboards approached it). But I mean also the use of various instruments in the increasingly large ensembles of the 18th and 19th centuries. We tend to obsess over temperaments in keyboard instruments mainly because there is the most discussion in historical treatises about how to temper keyboard instruments. But that's simply a practical concern that grew first out of organ tuning, as organs have long sustained tones and timbres with rich sets of harmonics that makes tuning choices particularly important. Later, temperaments became important to harpsichord and other keyboard players, as these instruments were some of the only ones that required the detailed knowledge of tuning to maintain them. A harpsichord needs to be frequently tuned, and unlike other instruments like strings (bowed or plucked) that typically only needed a few intervals tuned, a keyboardist needed to know how to create adequate relationships between all of the 12 chromatic pitches in the octave.
My point is that our perspective is biased toward keyboard instruments in tuning discussions because practically that's one case where knowledge of tuning was important historically.
However, if one considers the practical considerations of playing in large ensembles with baroque instruments, it's easy to see how tuning compromises would generally lead to something approximating equal temperament.
Baroque wind instruments, in particular, didn't have most of the valves and keys of modern orchestral instruments. A natural brass instrument without valves would be in-tune for some notes in its natural harmonic series, but other notes were often approximated through changes in embrochure and sometimes holes in the instrument that could be strategically used to better approximate chromatic notes or those not in the harmonic series of the fundamental of the instrument.
Meanwhile, woodwinds with few (or no) keys depended on all sorts of cross fingerings, half hole techniques, etc. to approximate chromatic notes. These tunings could be all over the map. Skilled wind players would learn to use different fingerings in different circumstances -- some fingerings might be higher in pitch (and appropriate for a leading tone), others might be in the middle of the chromatic range. Some might be better for softer or louder notes, but might be more or less out of tune.
Now imagine ensembles like those seen in Bach's Brandenburg Concertos and think about the challenges inherent in having all of those instruments trying to intermingle and be in tune with each other. You have brass pitched in one key (and in tune for most notes in that key), but woodwinds might be primarily pitched in another key and using chromatic fingerings attempting to be in tune with everyone else. And these all needed to play well with the strings (who had more flexibility in tuning their individual notes, but mostly tended to anchor to the perfect fifths in their tuning) as well as whatever keyboard in whatever temperament that was accompanying them.
If one reflects for a moment on the disaster inherent in trying to mesh all of these tuning considerations together, it's easy to see that the larger ensembles of the 18th and 19th centuries required gradual compromises to a standard tuning across the scale. All of the tuning debates about organ tunings and meantone and well-temperament fly out the window when you're just trying to have the flute, the oboe d'amore, the horns, and the strings sound okay when playing along with harpsichord in several keys.
So, as instrument builders started to invent and add keys and valves in the late 18th and early 19th centuries to wind instruments, the obvious choice was to approximate something like a compromise temperament where all the chromatic intervals of the scale were roughly equal. Players of these instruments were already doing this in large ensembles, out of necessity.
The expense of creating these more complex instruments increased, so gone were the days when a wind player might expect to have several instruments pitched in many different keys. Instead, the new holes and keys and valves were meant to make instruments useful in just about any key. Wind players today might choose to still have two instruments (usually one pitched in a flat key and one in a sharp key, like the B-flat vs. A clarinet) for convenience in fingering, but the new systems for chromatic scales on instruments were necessarily close to equal temperament simply to allow ensembles to play together reasonably in tune.
None of this takes away from the role that keyboard temperament theories and increasing chromaticism may have played in leading to equal temperament, but the simple fact of larger ensembles trying to play in tune in several keys was also likely to lead toward a compromise tuning for the chromatic scale anyway -- regardless of the piano or even the use of keyboard instruments generally.