The domain scenario makes management easier, but the idea of the domain
controller going down seems problematic.
Nope, totally not. It actually is a supported scenario that was introduced in one of the earlier versions, even for clustering (and THAT was problematic as the cluster could not start without an AD controller - now they can).
There is absolutely nothing on that that is not supported as standard configuration for years now.
I saw one case where the DC VM wasn't set to always start up automatically.
Some people are idiots. And not setting up a DC as autostart without a good reason is in that category. That is not an argument. I once saw a car burning out - that does not mean that it was not a technical issue.
Most small sites don't have the budget to buy a second physical server
Sorry, that is as bad an argument as it goes. Those are the same sites that whine like mad - the moment the first server fails, and hardware DOES fail. If your business depends on a server, how fast is second server more expensive than paying people or rent for a non-business? Same level of argument like "we do not do backups" - nothing to do with corporate (THOSE have plenty of spare) but with common sense.
I can either turn away their business or make some compromises to do the best
I can with what they can afford.
Turn away their business. Unless YOUR business depends on it, refusing to do reckless work has no implication for your financial health.
Is the above situation with the DC as a VM a serious issue?
I am sorry, but given it is a documented standard scenario.... that burns down to "how much do you know of machines that you work with". It once was a hack, it has since then been standardized.,
Could extended DC downtime create a situation like I mentioned where you
couldn't even log into the host in order to start or repair the DC?
If you are not careful enough and disable the local admin password which can serve as an emergency access point (and which should not ever be disabled on a scenario without plenty of backups) and that DOES exist the moment you do not run AD on the machine.... there is no risk.