I just did some searching, and I couldn't find a single reference to any law criminalizing drunk riding that isn't fairly recent.

Looking at some of the state laws criminalizing drunk riding: Minnesota only criminalized it in 2000, Oregon explicitly extended the motor vehicle code to riding animals, and it appears several other states(NH,SD,MA) it was the courts determined that horses were considered vehicles under the law. Interestingly SD explicitly excluded horses from the definition of vehicles back in 2006. I didn't look at every state, but from those laws and a few articles I found about the earliest drunk driving laws(this one about Vermont is interesting) it seems likely that the constable was wrong.

I think the reason drunk riding probably wasn't illegal before cars is that it didn't really need to be. Horses aren't machines, they don't do exactly what the rider wants them to do at all times. If a rider is completely wasted and passes another horse or pedestrian, the horse will instinctively avoid them even if the rider directs them toward collision. It really only becomes an issue when 2000+lb machines zooming around that have the potential to scare the horse that the rider needs to be fully capable of taking control.

I don't know why I just did an effort post about this, but I found it interesting.

All Constables Are Bad, never trust a word out of their mouths

It's intersting topic. Currently in Poland it's in the somewhat gray zone - you can lose licence and get heavy punishments for driving a vehicle when drunk, but horse isn't a vehicle according to the law definition (some paragraphs even specify "mechanical vehicle"). But in the road codex there are some exclusions where you can ride a horse or horse wagon, so you would need to pay attention to those. Also riding horse when drunk on the road would met the definition of "causing a danger in road traffic" and this is punishable too, though lighter than drunk drive.
Finally there is interesting loophole (afaik not checked yet) where a horse would go on autohorsepilot while the human would be lying drunk to total unconciousness on its back or in the wagon, so it would be unclear if he could be even counted as rider/driver.

Of course all this was nonsense in the relevant 70's/80's back then it was afaik legal and even if not, nobody i knew ever got into trouble for this. We also had maybe 1% of current cars so understandably the regulations were more relaxed and on such villages horse wagons and tractors were equally common sight as cars on the roads.