I've tried installing multiple third party apps in Ventura and had no problems doing so as the button to "Allow" always appeared in the Security & Privacy Settings. This even worked for third party apps which included drivers. I tried apps from developers that were not part of the Apple developer program.
Of course I have had problems over the years getting third party apps to work because macOS does not always display the "Allow" button. My guess is the developer is doing something wrong when developing their apps for macOS, but it may also be the quirky nature of macOS as I have seen a lot of macOS quirks.
Like @Barney-15E says, Conrtol-clicking or Right-clicking an app to open it the first time is the most reliable method of making sure the app gets approved by macOS Gatekeeper. If you still have a problem, reboot the Mac. If you still have a problem, then reboot the Mac in Safe Mode and try Control-clicking or Right-clicking the app to open it. You may still need to reboot the Mac and boot into Safe Mode a second time. If Safe Mode works, then it means you have some other third party software installed which is interfering with the normal operation of macOS such as anti-virus apps, cleaning/optimizer apps, or third party security software.
If you still cannot run the app, then you need to contact the app's developer for assistance and to make sure the app is compatible with Ventura. Also, Ventura as an update to 13.0.1 so make sure you have this update installed as well.