Sending all the useless horses to glue factories in that time was so prevalent it was a cartoon trope. The other trope being men living in flop houses, and towns having entire sections for unemployable people called skid row.
The AI people point to post 1950s style employment and say 'people recovered after industrial advance' and ignore the 1880s through the 1940s. We actually have zero idea if the buggy whip manufacturer ever recovered or just lasted a year in skid row before giving up completely, or lived through the 2 world wars spurred by mechanisation.
Horses were killed more often for meat that was used in dog food than for glue.
I did a deep research into the decline of horses and it was consistent with fewer births, not mass slaughter. The US Department of Agriculture has great records during this time, though they’re not fully digitized.