Hi Dmitry,
Hm-m-m...
AFAIK, REST API already exists in TestComplete/TestExecute and is used to drive TE from within TestLeft. It is just disabled in TC by default but it is no problem to enable and consume it (https://community.smartbear.com/t5/TestComplete-General-Discussions/How-to-provide-an-object-from-third-party-components-to/m-p/175381/highlight/true#M32657). Please correct me if I am missing something for the REST API been suggested.
On the other hand, it is OS requirement to have active not locked user session in order to be able to run UI tests. Unless it is possible to check for the active user session existence and provide it if necessary via suggested REST API, I hardly see the benefit of REST over the regular command line. (There is no problem to remotely login into test box and start test within remote desktop session using command line tools now if the account has proper permissions.) I guess, that REST API might be more convenient if tests are executed within a cloud (e.g. AWS), but not sure at the moment about licensing.
I would agree with baxatob that it would be convenient to be able to get information about remote active test run without the necessity to attach to the remote session. (Attaching itself is not a problem, but it is not convenient to switch remote session back to console.)
However, I think that it may be even more useful and convenient if TestComplete provides some listener component that can be installed standalone. For example, on the Jenkins or some other CI server. Then, when needed, running remote instances of TestComplete/TestExecute can use something like push notifications to feed this listener with the info defined by test developer. (I am not sure that it is a good idea to push all test log to the listener. As a possible option, Indicator object can be enchanced with additional parameter to push its contents to the listener when pushing functionality is enabled for the test.)
This listener component can be used than to provide an overview of the active test runs (grouped by test machines) to console, CI server panel, messaging service, etc.