It is mystery for me, why doesn't DLSR/System cameras doesn't store this information in video and photos. It can be useful for both - for video stabilization and for photos, taken with long exposure times, de-convolution... It is very simple technical solution, but no, nothing from Canon, Nikon, Fuji...

You have a good point and this does seem to be evolving industry practice. Sony does this in a few of their recent cameras. And Blackmagic just released a new firmware for certain Blackmagic Pocket Cinema Cameras where gyro data is stored in BRAW and then Davinci Resolve has gyro-based stabilisation. It’s not perfect, it’s not magic, it’s not a substitute for a gimbal, and there are some real limitations but that aside the results are shocking. My understanding is that rotations can be very well corrected (as it’s same camera position) but translations are not as successful. Also you typically need to shoot with higher frame rates and there is a significant crop on the resulting image. All that being said I use this frequently and my handheld casual shots are much much nicer now.

Is there an app you'd recommend to capture that motion data?

I heard about this at one point, by was stymied by not knowing what I could use to record that data in a useful format for gyroflow.

Yes, the compatible apps are listed on their github:

https://github.com/gyroflow/gyroflow#supported-gyro-sources

There are a few apps listed at the very bottom of the link: “Mobile apps: Sensor Logger, G-Field Recorder, Gyro”