No, not quite. 60fps (or 50fps for PAL regions) was basically always the target for 2D games rendered at 240p, because it was easily achievable with the hardware and looked way better.

When 3D games started to appear with the release of the PS1, Saturn and N64, 60fps was often out of the question because the hardware wasn’t powerful enough.

Interlacing doesn’t really have anything to with that. Interlacing just means that of the 480 lines of resolution, only half get shown on screen every frame (always the even or uneven ones). Since the image gets rendered in 480p anyway, it doesn’t matter whether or not the game gets rendered at 60fps or not.

Except when using field rendering, which was popular on the PS2. There, only the lines that get shown that frame get rendered, which is why it only works at 60fps. Maybe that was what you were thinking about?