I want to use my own machines to build certain jobs (perhaps because those jobs require heavy computation), but I want to use the free, shared, cloud-based runners for the rest (to increase overall throughput). And I want to do this while keeping the master CI instance (the node which keeps track of which jobs are running and serves the webpage which shows the job status) on hosted infrastructure so that I don't have to manage it myself.

Nearly everything that exists today, except Gitlab, forces you to choose between shared runners in the cloud and doing it completely yourself (including hosting the master CI instance). Gitlab allows you to connect both shared and private runners to a hosted CI instance, so we get the benefit of private machines, the throughput of shared machines, and ease-of-use of not having to host the CI ourselves.

Edit: And note, to use private runners with basically any of the commercial CI's, you have to pay big money for a enterprise instance (even if you're open source). Whereas the ability to connect private runners to Gitlab comes for free (I think on all accounts? but definitely open source ones).

I think the latest features we launched at Shippable meets all of your requirements. You can mix and match on-demand and dedicated nodes, allocate them to specific projects if you'd like, all of this is available in our SaaS edition, and pricing is (in my opinion) quite reasonable.