I have a hobby android app idea that basically just uses Android's sensors and logs them long term for several days (external battery). The sensors needed are in $200 phones, and I can get ones for under $100 if the screen doesnt work.
My question is, is it pretty easy to make an app that starts and loads via debugging, keeps running when disconnected and when I connect I can extract the log file, or would this so be difficult I would be ahead just to spend the extra $100?
As mentioned, turning on USB debugging without the screen isn't possible. You can't enable USB debugging over USB for security reasons, so your only option would be to use hardware commands to put the device in firmware download mode (presumably it will have a way to do that), then load a custom OS which allows USB debugging by default.
Personally I think that's more than $100 worth of work, so I'd just go with the working device. Then you can use it for other stuff down the line.
Alternately, you can probably get a replacement screen for not too much, and for most devices changing the screen is relatively easy. I'd look into that option as well.