There were several strategies for dealing with the problem of ice resurfacing before Frank Zamboni came along.
Just Let the Ice Get Bad:
Before 1910, hockey games were played in two 30-minute halves. ESPN says "By the end of each half, the ice was full of ruts and covered in snow, and the game slowed to a walk." So eventually they switched to three 20-minute periods, allowing an extra opportunity to clean the ice, even though this severely slowed down the pace of the game. Even today, NHL hockey has longer stoppages of play than other professional sports, with two 18-minute intermissions as compared to a 12 minute halftime in NFL football and a 15 minute halftime in NBA basketball and FIFA Football. (I won't try to quantify how much downtime baseball has.)
Cheap Labor: To resurface a rank, you need to clear the snow. This could be done by handing out a lot of shovels. For example, at the UND Barn, "Young fans (in exchange for game admission) prepared for resurfacing by shoveling shavings and snow off the ice from the preceding period's action." Even in our post-Zamboni world, many NHL teams use teams of young women, so poorly paid that they can't afford winter clothes, to clean the ice between periods.
Other Contraptions: Tractors with ice scrapers were used in some rinks. After shavings were cleared, Zamboni (and presumably other low-budget rink operators) used what was basically a wheelbarrow and a hose to flood the ice. The more mechanically-minded managed to come up with better flooders. Returning to UND:
. . . a between-period entertainment (for those who weren't forced by the
cold into the warming rooms) was watching him artfully resurface the
ice with a pre-Zamboni apparatus he had cobbled together and which was
pulled across the ice. It consisted of two barrels, one welded on top
of the other, pipes, and valves that directed hot water to the ice
through a canvas strip at the bottom rear of the Rube Goldberg
contraption ("My barrel flooding outfit," Purpur called it).
Here's a Canadian Parks and Rec department hand flooder:
Zamboni first tried to outfit tractors with mechanical sleds, before realizing that he could combine the shaving, cleaning, and flooding machines into one by building off of war-surplus jeep chassis:
Clearly Zamboni's invention was effective enough that now nearly all rinks employ one. But there was enough amateur mechanical ingenuity in the first half of the century that I wouldn't be surprised if there were rink managers other than Zamboni and Purpur who were experimenting with their own shoestring resurfacers.