As you have noted, the Run-time edition of Test Studio does not have any built-in mechanism to be updated, unless installing manually the latest version. The only thing I can add on this topic is that you do not need to uninstall the previous version, as it will be automatically upgraded to the latest you are installing.
Apart from that you can review the Test Studio product versions installed on the machine(s), which host(s) the Scheduling and Execution services from the Execution Status view. The version of the Execution client can be found on each separate Execution machine.
I remain available to continue the discussion if you have any concerns or ideas to share on the topic. Thank you for your cooperation in advance.