IMHO, that is one of the dumbest mechanics' myths out there. I'd have asked for a Honda TSB that recommends not changing the transmission fluid in older transmissions. There isn't any such thing. Instead, a 3x or 4x drain and fill would be best to change out as much of the fluid as possible if it does not look to be the right color, smells burnt, or has had an unknown service history.
They were pulling this same thing back in the 70s and 80s. I was driving a '73 Catalina (which our family owned since 1975), and since we'd never had the transmission fluid changed, we decided to have it done. The grease monkey insisted that the crud inside the transmission was "holding it together." I hate to tell that grease monkey, but we never had a problem with the transmission for the remaining few years we owned it. Shifted as good as it always had.
Unless the Fit had been neglected prior in life (provided you bought it used), something else could have happened to the transmission. "Burned up inside" doesn't say a thing as to what component inside the transmission actually failed.
I changed the fluid in our '88 Accord and had it fail--it was just a poorly designed transmission. And ask thousands of Acura TL owners from 2000-2003 who had theirs fail, some multiple times, prior to the first scheduled fluid change. If I remember, it was the clutch packs between either 2nd or 3rd gear that would burn up due to a lack of lubrication. The only true "cure" was to get an automatic transmission from a 2007 Accord V6, as it was a totally new transmission design, vs. rebuilding the poorly designed transmission that could easily fail.)
The Accord in the family has a transmission issue, but the trouble is finding a trustworthy shop to look at it. Too many just say "rebuild" without even looking at it.