I think the problem is that rainbow shows banding, but only in some arbitrary narrow range, and then pretty much ignores a large swath of values that all map to

I think the problem is that rainbow shows banding, but only in some arbitrary narrow range, and then pretty much ignores a large swath of values that all map to slightly different hues of green. So the "bands" you see are not representative of the underlying data, unless you're lucky. If you want to see bands I think you're better served by just digitizing the value into, say, 10 ranges, and color each range in uniform color, like how they did it in old-school paper maps.