You actually can press CTRL+SHIFT+i to open dev tools in Postman. It is just electron after all. Also even if it were some other UI toolkit you could view the network requests via Fiddler/Wireshark/etc.
FYI, only requests made with the XHR/Fetch APIs in Electron apps show up in the console. Insomnia (I am the maintainer) uses libcurl to send requests, which do not appear in the console. The NodeJS http/https APIs also do not.