iPhone 5s won't turn on, flashing logo, no restore, but WILL reach DFU

My iPhone5s is obviously old and its performance and battery have gotten bad. Now the phone has shut off and will not respond. Nothing at all occurs when the phone is unplugged.

When plugged in, the Apple logo cycles on and off, and occasionally there is a red screen momentarily in the mix. I cannot get the phone to turn on and cannot get it to go into restore mode. However, I CAN get it to go into DFU mode when I hold the power button for 3 seconds, while still holding the power button press the home button for 10 seconds, the release the power button while still holding the home button until iTunes recognizes the recovery mode.

I would like to assume if the phone is working enough to get to recovery, it can still be accessed to not lose all the data - is that possible? As always happens this phone hasn’t been backed up for a long time.

Any ideas on how else I can get the phone to a point of accessing the data to transfer off? I’ve looked through other solutions on here but haven’t found anything that worked.