I think it's because the MMC3 did not yet exist, and the MMC1 wasn't good enough for Punch Out.
The MMC2's main feature is that it can automatically bankswitch whenever it encounters a particular tile (tile FD to switch, then tile FE to switch back).
The game often surrounds text with FD tiles, then uses FE tiles after the text.
The MMC2 has a CHR bank size of 4K. So in order to use both the CHR bank which holds the font, and another CHR bank which holds portrait graphics, it uses this method of automatically switching CHR ROM.
In this game, FD and FE tiles are all blank, they look like a space, but cause the MMC2 to trigger CHR bankswitches.
Examples:
The title screen surrounds the text with FD tiles, and follows the text with FE tiles so it switches back.
The pre-fight screen surrounds the opponent's portrait with FE tiles, then uses FD tiles to return to the graphics for the font and Little Mac portrait graphics.
The fight screen uses FD tiles to select one set of graphics for the top of the screen. Then it switches to another bank of graphics for the bottom half. Every single empty tile below the top part is actually an FE tile, I didn't want to bother annotating them all.
The FD tiles for the top half are actually off the screen, but due to how the NES works, it fetches some extra graphics when drawing its picture, so those trigger a CHR bankswitch anyway.
The auto-bankswitching feature can also be used for sprites. The top half of the boxer could be on one CHR bank, and the bottom half could be on another CHR bank. If there's a FD or FE sprite, it will trigger a CHR bankswitch. I don't know offhand of any animation frames that use two different CHR pages, so I can't think of any illustrations.
Now for the question of why they had to use such an odd way to do graphics switching... They were using 4K CHR bank sizes. If they had used smaller bank sizes, they could have switched graphics in smaller size chunks, and not needed all that circuitry where it watches which tiles get drawn. I suspect that Punch Out could have worked just fine if it was designed for an MMC3.
There are a couple other games that also use a mapper similar to Punch Out: Famicom Wars and Fire Emblem. Except they switch PRG differently, but are otherwise the same thing as the Punch Out mapper.