Okay...that really sounds like it's SSDT then. The execution log views won't do you any good - they are really more for the report execution.
You can try looking in the Reporting Services trace log - that one I mentioned the other day. Should be in the following on whatever drive where you installed:
Program Files\Microsoft SQL Server\MSRS13.MSSQLSERVER\Reporting Services\LogFiles
Those files in there are what are often called the Reporting Services logs. MS often refers to them as the trace logs. Anyway...with SSDT it's often VS shell that can cause issues or in configuration manager, things are messed up. I was reading about a lot of problems and known bugs with one of the recent versions. It's like they all have different bugs...irritating as H. I recently just uninstalled mine as it was pissing me off too much. Here are a couple of deploy issues for one of the buggier versions:
Unable to deploy SSRS report
SSDT - Deploy Report To RS Gives "The given key was not present in the dictionary."

I would try downloading another version - that's what I was doing when I hit bugs. The bugs almost seem like they change on different releases. So you learn to live with the ones that don't mess up your work. Those ones usually if you restart you are fine. But a problem with deploying isn't one to live with.
Are you getting any specific errors when it won't deploy?

Sue