Your requirements pretty well match the Amiga OCS, which was introduced with the Amiga 1000 in 1985 and widely popularized by the Amiga 500 in 1987. So by 1988, that level of graphics was already common in home computers.
Actually, the Amiga chipset might not be a bad target for your project. Good Amiga emulators like UAE are easily available, there a plenty of good native development tools and tutorials still around if you search a bit (Aminet has just about everything you could need; the challenge is knowing what to look for in that giant haystack), and once you're ready to move beyond emulators, real Amiga hardware is still quite practically obtainable. And sticking a plain off-the-shelf Amiga box into an arcade cabinet would not have been an altogether unprecedented move in 1988.
Anyway, here's how the OCS compares with your requirements:
Composite output (possibly encoded from RGB with an external chip), X resolution 256 px, Y resolution between 192 and 240 px (more than 224 lines would be hidden by NTSC overscan anyway I guess)
Check. NTSC output at a nominal 320×200 px or PAL at 320×256 px, more with overscan. Vertical resolution could be doubled via interlacing, horizontal resolution could also be doubled at the cost of reduced color palette and sprites.
Palette: At least 32 colours simultaneously - (if either sprites or BG graphics are in 4BP it's enough to have a single palette (Sega Master System style))
Check. Up to 6BP in lores mode (320 px horizontal resolution) or 4BP in hires mode (640 px), with 32 independent 12-bit color registers. Full 12-bit color space could be shown using a special 6BP "hold and modify" graphics mode, although with restrictions that made it rather impractical in games.
The display-synced Copper co-processor also allowed easy mid-screen palette changes, extending the available colors, and could even be used to split the screen vertically into multiple parts with separate palettes, frame buffers, sprites and even different graphics modes.
A lot of 3BP or 4BP sprites, at least 16x16 in size (if larger is available it doesn't hurt) (if there's enough sprites available then only one background layer is necessary)
Kind of. The Amiga chipset had only eight 16 px wide 2BP hardware sprites (which could be merged into four 4BP sprites), which might not count as "a lot" for you. However, you could easily multiplex them with the help of the Copper or just use blitter objects instead. So in practice, the real limitation was 8/4 sprites visible per scanline (possibly more with mid-scanline multiplexing), plus as many bobs as you could draw in a frame (which was quite a few, especially if they were small).
One fully scrollable background layer capable of 3BP or 4BP graphics with either 8x8 or 16x16 character tiles.
Possibly one simpler non-scrollable background layer of 2BP graphics that could serve for backdrops or text overlays
Yes, kind of. The OCS had hardware support for two independent 3BP layers ("dual playfield mode"), so you could have simple hardware parallax scrolling at the cost of reduced colors. However, in practice, I believe most games would just use the blitter to draw all layers into a single 4/5/6BP raster buffer, allowing more colors and a potentially unlimited number of layers. As noted above, things like top/bottom status bars could also be done as separate subscreens with the copper.
Ps. Here's a random YouTube video I found by Googling for "best looking Amiga games" that shows off some examples of what the Amiga chipset is/was capable of. (I believe most of those are OCS compatible, although I haven't really checked. Not a lot of Amiga games really made use of the extra capabilities of the later AGA chipset even after it was introduced in 1992.)
