The procedure of first testing updates in a staging system and only then importing them into the live system, and the whole thing at regular intervals, is a professional procedure which I consider absolutely advisable for projects like yours, which according to you is “critical”. Why the procedure in your case takes 3 hours per run is difficult to say if you don’t know the project. There are various factors that influence the time.
For example, the amount of data. If the project is several gigabytes in size and the hosting used is relatively slow, each import of changes from Live into Dev and vice versa can take some time – especially if you really synchronise everything (including media files). For projects that are smaller in terms of data volume, this is of course faster.
Another factor is the possible requirement to test updates extensively first. To do this, you have to work through a checklist in the staging system that covers as many functions of the website as possible. Depending on the complexity of the website, this process can take more or less time.
Another factor is the way in which staging and live are synchronised in your project by the agency. For example, if they use a git repository as a basis, this must also be supplied with the updates. It must be documented which branch, which tag, which functional scope. This can also take time.
How quickly the whole thing is done therefore depends very much on your requirements. If you really want to make sure that the project never breaks down, then you should continue to go the way the supervising agency has gone for you.
Alternatively, you could also:
a) carry out pending updates directly in the live system. In doing so, you run the risk of malfunctions and even the complete failure of the project. However, you would save a lot of effort (and obviously also money) because it is done quickly.
b) Or at least do without a checklist. So just install updates in the staging, briefly check whether the staging can at least be called up, and then install the same updates in the live system. This should also be much quicker to do.
You will hardly be able to argue against the fees charged by the agency for their work. That is their business decision and has nothing to do with WordPress per se. But perhaps you can save time with the above thoughts.