Edit: as sibling comment noted, Paperboy does have custom logic on the cartridge, a 74HC161 4-bit counter. I think this is just used to switch between CHR banks.
This wasn't a big problem, because TVs of the time often had big overscan areas (areas that were rendered, but covered by the TV's physical bezel). However, it was visible on many TVs, and today it is very visible under emulation. Activision's programmers found this unacceptable. Rather than perform the very difficult, or even impossible, task of making their code run faster than the HBLANK period, they instead chose to render the first cm or so of each scanline intentionally black! Notice the width of the screen in this screenshot compared to other nearby games: https://videogamecritic.com/2600ff.htm#rev203
More crazy tricks like this are described in the excellent "Racing the Beam" book by Nick Montfort and Ian Bogost.
For the NES, depending on the registers in question, it can be easy or hard to update them during horizontal blank. Changing the scroll registers is easy, the PPU was designed to make that possible, and gave you a sprite zero hit test so you could get the timing correct--which is how Super Mario Bros. draws the HUD at the top of the screen.
With some extra hardware you can make this easier and get some cool effects like parallax. The NES exposes a NMI line to the 6502 on the cartridge connector, and you can wire up some logic on the cartridge to signal NMI for every scanline. Battletoads uses this effectively. Atari 2600 lacks NMI because the 6507 doesn't have a pin for it.
Certain registers are much more difficult to change during horizontal blank, but not impossible. For example, palette entries. This requires very precise timing so it was very rare to see it. Indiana Jones and the Last Crusade changes the palette entries mid-frame, but only for the title screen.
This was a workaround to avoid Atari patents. On the 2600 you could write to a register to halt the CPU until horizontal blanking (commonly written as sta WSYNC), to get perfectly synchronized to the next scanline. Sprite zero is more flexible, but the timing isn't as precise.
Some of this may have been undefined behavior, but the MMC3 chipset involved was produced and manufactured by Nintendo. Whatever the original design was, Nintendo appears to have stuck with that same set of capabilities throughout the console's lifetime, and encouraged their developers to take advantage of all the tricks.
I think there are games out there that do use sprites to do the same on the other side.