Lead-out grooves were already in place at Columbia and Victor by the mid-1920's.

the main reason the label got smaller was to make room for the eccentric ending groove to trip changers. The jukebox industry got the record companies to standardize on an eccentric groove in the early 30's. Most early changers and jukeboxes used an eccentric trip that activated from the reverse motion of the tonearm swing once it got to the finishing groove. Victor had started that in the early to mid 20's when they added a mechanism to automatically stop wind up victrolas, but many small labels did not add the grooves until Jukeboxes became popular.
When Lp's came out, the lighter pickups often bounced out of the groove due to the high forces required to operate the eccentric trip ratchet, so then changers went to something called a velocity trip, and the eccentric groove was eliminated on lp records. 45s never had eccentric grooves, that was another reason for abandoning the eccentric trip.

I should correct you on that last point on eccentric "trip" tracks on 45s, The W. & G.-processed "Golden Fleece Top Hits" 45s(and occasional 7" 33 &1/3 EPs) did indeed have the eccentric run-out "trip" track. Just thought I should point that out.