Short answer: your power supply should work - you just need the right voltage, current rating, polarity, and connector.
An unregulated power supply is a kind of power supply whose output voltage varies with current draw. In other words, it does not put out a constant voltage - it is only rated to supply a minimum voltage at a certain current draw. In this case, you will need a power supply that can provide 9V DC at 700mA. Because most devices require a constant voltage, you can always use a higher rated supply.
If you test an unregulated power supply with a multimeter, you will likely see the voltage is higher than what is on the label, say 9.5V DC. That is because the multimeter is not putting a heavy load on the power supply. As the current draw increases, the supply voltage will drop. That is why you can use a higher rated device, but not a lower one.
In your post you said it uses a 3.7mm plug, but the manual link says 3.5mm with a positive tip. I assume that it means it uses a 3.5mm phone connector for power, like what you would see on a pair of wired headphones. The important part is that the positive 9V is connected to the tip, and ground on the sleeve. Sometimes, reversing these connections can damage devices, but there is usually a protection circuit inside that prevents this, and I would wager that this is no exception.
PiKVM or a similar device could work for OP - is that what you are thinking of? I’ve used it and it works well.
I think a lot of people who self-host get caught up in the excitement of getting the services up and running and neglect disaster planning, prevention, and recovery (myself included). Either they put it off for later or don’t realize it could be a problem down the road until it happens. We always say not to self host anything you can’t live without, and most take that advice, others don’t. Not saying OP falls in either category, necessarily, just adding on to some of your points.
Self hosting really is the land of compromise where we all have to balance our requirements, budget, time and effort. Personally, I have a little disposable income that I spend on hardware to host non-critical services so I can learn and tinker. It could all go away and all I will have lost is the time and money I put into it, but I gained some knowledge and enjoyment. Needless to say, I don’t have much in the way of backups and monitoring.