Mauna Kea not Everest of cyclists
The one thing that's great in my mind about Mauna Kea is that it starts from sea-level, but in terms of altitude its summit is lower than Mount Evans', a really great climb that one with great scenery (much much nicer than Pikes Peak) and a nice race held every year.
As for paved roads, La Cumbre, just outside of La Paz (Bolivia), on the road to Yungas, culminates at 4653 meters, I have climbed it several times. There was a race up to the pass in 1966 (probably also on previous years) but I missed it, at the time only the first few km were paved and at places you sunk deep in dust. In 1997 it was paved to the top and a bit further.
As for unpaved roads, you have a number of them in South America (Peru, Bolivia in particular) above 5000m. I have gone up to Mount Chacaltaya a number of times, starting at various altitudes (3200, 3500, 4800m). The road ends at 5240 m. at the ski lodge (no more skiing, thanks global warming) after passing the entrance of the Cosmic ray lab at 5220m.
In the 1960's when I first climbed it, the road was horrendous and my gearing on my transformed racing bike (fat tires) totally inadequate. The road was almost as bad the last time I climbed it, in 1997, but in spite of my 30 extra years it was much easier (but a bit slower) as I used a mountain bike. I understand the road was somewhat improved a few years ago.
You could even ride up Uturuncu in South Lipez at 6000m without seeing any snow but I understand there is a stretch in fine ash which seems unrideable by mountain bike.
Sillayajuay is another 6000m. volcano with a road to the top, but occasionnally it has snow. (That's what a Bolivian guide told me last year). I found Sillayajuay on google but spelled a bit differently. Never heard of a cyclist going up there.
In Tibet and ****stan-China (near their border in particular) there are a few passes above 5000 meters, but the altitude often quoted for Khardung La is really "only" about 5200-5300m. I won't repeat here the erroneous altitude you usually read.
Looks like ****stan is a dirty word

Let's try Thecountywhereoussamadiedstan"