Hi Zack,
A continuous stream of reboots every three minutes is the distinctive characteristic of what's called a kernel panic. When heart of the iOS operating system, or "kernel" detects an unrecoverable error it generates a system panic that logs the cause of the panic then reboots the phone.
In your case, there's a hardware fault that's being detected in the phone's sensor sweep, which just so happens to occur every three minutes. The kernel sees an unrecoverable error; i.e., it can't read a sensor, and so generates a kernel panic.
There are files on the phone that contain the information you would need to diagnose the problem, and it may be possible to retrieve those files even without a screen using external tools and a PC. The other thing you can do is to go ahead and replace the screen then if it's still boot looping you should have enough time to locate and download or screenshot the panic log. Here's our "bible" showing you how to obtain and understand those panic logs.
[[iPhone Kernel Panics|iPhone Kernel Panics - iFixit|new_window=true]]
As mentioned in the Wiki article, you can download the iDevice Panic Log Analyzer to transfer the log to a computer, assuming you can get them to talk without a screen.
[link|https://github.com/waynebonc/iDeviceLogAnalyzer-public?tab=readme-ov-file|GitHub - waynebonc/iDeviceLogAnalyzer-public: A quick and easy panic log extraction and analysis tool for iDevices.|new_window=true]
If you can grab the latest panic log file and add the first page of it to your question, we can help diagnose what's causing the reboot.
Keep in mind that due to Apple's repair-antagonistic policies, you will get "genuine parts" warnings for both the display and the battery unless you pay Apple to replace those parts. It is possible to get around those warnings for the battery, but the display requires precision microsoldering skills in order to keep from getting those warnings. Without the workaround for the battery, you'll also lose access to the battery health information, and lose the True Tone function on the display without using a device programmer.