Can badly cyan-shifted JPEGs with clipped channels be color-corrected?

Probably not to accurate color. Based on the examples and histograms, this looks worse than a normal white-balance error: multiple channels appear severely clipped/posterized, so much of the original color information is simply gone.

What you can still do:

• Try a basic rescue in a RAW/JPEG editor by reducing vibrance/saturation and adjusting white balance and color balance.
• If the result is still unnatural, a monochrome conversion will usually work better than forcing color.
• Since the red channel appears least damaged, using mostly or only the red channel for a black-and-white conversion can give the cleanest result.
• After that, use curves/contrast and mild sharpening for presentation.
• If pure B&W feels too harsh, a toned or “tinted postcard” style can preserve some mood without pretending the color is accurate.

In short: with clipped JPEG channels, there usually isn’t enough valid color data left for a true restoration. The best-looking output is likely a strong B&W or gentle tinted treatment rather than a faithful color correction.