I would imagine such a person would heavily script a lot of things and be able to spread their time among hundreds of customers, and I'd get the cheap rates if I'm willing to pick off their menu of preferred software packages for each need, rather than picking my own specific one, for instance.
I don't know if it's realistic, but I'd love something on the order of $20/mo for something simple like my own photo gallery (whatever package they recommend), maybe a peertube instance, a personal pastebin or etherpad. I'll provide the hardware and connectivity, I just want someone to help me set it up, log in periodically and apply patches, and manage backups and stuff.
I'd expect to pay more if I had weird requests or high-maintenance stuff like email.
Some time ago, I heard an interesting idea for "investing" in indie musicians -- buying their music actually buys stock, and if they become popular, your shares become more valuable. Similarly, if I'm picky about wanting my CTO-on-demand to support _this particular_ photo gallery software, and I pay extra for that special service initially, but later it becomes popular and many of their customers request it and they're able to service those requests because I paid them to develop the skill, maybe I get something in return? I dunno, that gets complicated, but it's worth thinking about how the incentives work.
What would your ideal budget range for this role be? Maybe we can find a 5-star person, share out their time among 4-5 people here on HN…