Also, does this rule out hosting on clouds that don't offer vTPM support? (Are there any that do?)
Chrome OS uses TPM heavily[1], and iOS has the Security Enclave. The standard TPM API is PKCS#11, so any hardware that speaks it can be used with any software that speaks it.
Problem with TPM is that the whole hardware and software stack needs to be secure, which in practice means it needs to be designed top-down with awareness of the TPM, and audited. The secrets must not be cached, written to file system, kept in memory, leaked over network. There are implementations such as Trousers[2], but it's more or less just a proof of concept; it may provide additional security, but most likely you're just using a very complex lock, and leaving the key under the mat.
[1] https://www.chromium.org/developers/design-documents/tpm-usa... [2] http://trousers.sourceforge.net/man/tpmtoken_setpasswd.1.htm...