iPhone 6s 16GB (iOS 13) stuck in boot loop.

Okay this is a sticky one. My iPhone 6s 16GB running iOS 13 is stuck in a boot loop. This one is different from the usual ones because this did not happen during a software update or anything to do with software corruption. I had a notification badge on the Messages app with seemingly no new message. When I backed out of the app I noticed that the phone was laggy. Now I know that this may raise red flags about this having something to do with the Messages app (bug or corruption) but I think it doesn't. I should also mention that the iPhone was quite beat up (no physical damage) as it was quite old, so this random lag was not particularly unusual. Anyway, as an impulsive reaction, I decided to restart the phone thinking that should fix the notification badge and lag issue. I had done this hundreds of times before. And that's it! That's how the phone bricked. Now, to clarify what I meant with "beat up"....

The phone storage was almost full (couple hundred megs were free from what I remember). And the battery was bad. Really Bad. I did not care though because this was a secondary phone. the battery could barely last an hour of actual use. I tried all of the typical troubleshooting tips for this kind of issue. I tried going into recovery mode and updating my iPhone on iTunes (where it gives the option to update or restore). iTunes said that the phone could not be updated but only restored :( I know restoring may fix the issue but I really don't want to lose my data (and I did not back up my data for a long time before this, I know, really irresponsible of me :( ). Also, I only have the basic 5 GB of iCloud storage space which used to back up my data but that got over pretty quickly, so again, I will lose all of my recent data.

Naturally, knowing how bad the battery is, I assumed that one possible cause for the boot loop was that the battery is unable to output enough power for the phone to start up. I tried turning it on with a charger (Original Apple Cable and Wall Adapter) and also after draining and the fully charging it overnight. But all of that did nothing. I decided that I'm running out of options and I should bring it in at an Apple Authorized Service Center. They said that in order to check any issues with hardware, they must first rule out software issues. This meant that they would need to restore my device causing me to lose all of my data. When I tried to explain this to him, he said that this was the protocol and that he could not troubleshoot the hardware before restoring. So that didn't work. I then took it to a 3rd party repair shop, where the guy tried switching my iPhone on with another iPhone 6s battery and to my surprise, it still did not start! So I guess battery is ruled out. He also mentioned that the cause could very well be that the storage was full. He assumed that on his own, which makes me think that that's the actual cause. I should also mention that my storage was full for a very long time before this, and was also worse at times with it showing that the phone had only a couple of MBs left. When the issue began, as far as I can tell the storage was at least a 100 MBs free. So what could the issue be?

At this point I am running out of options and it seems like I may need to restore my device and lose my data :( oh and I should also describe the boot loop. What happens is that after holding down the power button, the Apple logo shows up and stays up for 1-2 mins before the screen goes black and the process starts again. Can anyone figure this out?