I'm not sure my data is being held hostage moving to Azure. It's not onsite now, and hasn't been for well over a decade. Plenty of service providers could have held something hostage, and still could. I'd argue it's less likely Microsoft would/could cut off access than some small provider. We would still do backups to remote places.
Could they change the cost? Sure. Same as any other contract provider we use. However, we can contract as well, and plan for rates.
Security is always an issue, but that's mostly our issue for now. We run on VMs and we're responsible. Azure or AWS provides some more protection from DDOS and other attacks because they scan traffic. We didn't do that in our own cage, and not sure what Redgate would do, but not sure Rackspace does anything here. They might.
These days, many, perhaps most, companies use some sort of "cloud" vendor. They use a hosting provider. I'd guess that many companies don't host all their own systems, perhaps just the public ones. They negotiate contracts for their hosting, just as they would for the lines that provide service to the office, to the owners of space, etc. The cloud can be month to month, but it can also be year to year. Most people that use the cloud go month to month because prices keep falling.
In the mainframe days, plenty of people were at the mercy of chargers from vendors as well. however, vendors need to do business, not squeeze as much out of their customers as possible. They value stability as much as the customers. I would say the idea that some cloud provider might hold a small company (like Redgate) hostage from their data is a bit over the top.
Is there risk? Sure. Moving anywhere involves risk. Rackspace is a safe choice, as I'd say AWS is as well. However, there's a cost there. Could we move to some other hoster? Sure. Risk there. Move back to the Redgate office? more risk, given the fluctuating Internet speeds I see over there. There's always some level of risk, but I think Azure/AWS is less risk than most.