On a personal level, I use 6 trusted apps that utilize screen and system audio recording and I expect that when I allow them to do so, that I am not prompted weekly or upon every startup.
On a professional level, it's just bad for developers when there is simply no work around to this.
Apple, please add an "always allow" option in the prompt. Thank you.
https://9to5mac.com/2024/08/06/macos-sequoia-screen-recording-privacy-prompt/
Have you raised a Feedback report? If not, please do so at https://www.apple.com/feedback/ then post the FB number here.