Yes I see the plane, but you need to think how you connect to it.
Vias are too far from the ground pads to be of much use.
This may or may not be the source of these particular issues, but I do guarantee you from experience that this layout is broken. But, the good news: it is easy to fix. Just add vias near the ground pins, move the bypass caps closer to the power pins (moving to a smaller capacitor package is highly recommend, even though not mandatory - even if you didn't like soldering 0402, just a few capacitors is not a daunting task to do!), and add vias there as well. If you already have the PCBs made, this can be fixed by drilling a few holes and soldering some wire.
Fixing the obvious source of problems
first removes one
very probable source of really mysterious problems later on. This particular issue either gets fixed, or does not. In the latter case, you can go on with other possible causes...
One simple way to do power distribution in 2 layers is to dedicate the space under the chip for Vdd. This way, the backside is free for complete ground layer, as you have now, and you don't need to jump between the layers as much. An example is attached where I did this. Well it's a 4-layer so the bottom
is noncontinous as it doesn't matter here because of the other layers, but you can see you could route the "power in" between the corner pins on the top layer for a similar 2-layer design.
It's not super critical, but when I was a beginner, it took me at least 5 years to admit that yes, the layout really might have been behind my random resets and reliability issues, since the problems just vanished once I started to listen to others and stop being stubborn with "it can't matter that much" attitude
.