I doubt your app is crashing. I also doubt you are going to find any meaningful information in the logs. And I'm not sure what you mean by "looking at console logs displayed". That suggests you might be unaware of significant changes to the macOS logging infrastructure over the past few years. You can't use Console.app like you would have years ago. It has to be up and running before you do your test. Otherwise, you have to use the new "log" command line tool as well as its predicate language for searching for relevent log entries in the past. It's not a trivial task.
When doing any kind of testing on Gatekeeper or other system-level interactions, I strongly recommend testing in a VM. A decent VM will support snapshots, making it easy to roll back to an inital state and test again.
I'm not going to take the same diplomatic approach as eskimo. Qt is just a dead end on the Mac - end of story. If you want to keep using it, fine. Please see my earlier suggestions on how to do an end-around of Gatekeeper.
That code signing documentation was first written in 2008 and has not been updated in over two years. In particular, it doesn't say anything about Notarization. About six months ago, Apple announced that Notarization was going to be mandatory. The other day, Apple finally announced when Notarization was going to be required and a number of popular developers are freaking out over it today. Or, if nothing else, they are enjoying a noticeable uptick in blog traffic 🙂.
But my point is that it is going to be difficult to beat your old Qt app into conformance with modern, and constantly evolving, security practices and infastructures. In my opinion, I don't think it is worth that level of effort. It would better to just identify the bottleneck, Gatekeeper, and change your processes to avoid it. That's an easy, 5-minute fix.
But if you do plan to distribute this app publicly, then a more thorough review is in order. Stripping out Qt would be a high-cost, but low-risk solution. Beating code signing into submission would be a moderate-cost, moderate-risk solution. Which is best for you? What other value, if any, does Qt bring to you? What other value would an alternative, but more mainstream approach, bring?