It seems this unsolved mystery didn't sit quite right with me. Here's a solver I wrote in python, that does not apply linear algebra techniques. I'll try to explain it here shortly, but if you want implementation details see the paste below. Although scaling severely worse than a linear algebra solution, the approach is capable of solving 7x7 grids. I would upload a video of a demo, but it was a bit of hassle to do with itch.
https://paste.myst.rs/1eux4vll
I'll use the term move here to denote the clicking of a tile. Firstly, iterate through all possible sets of moves that could be applied to the top row, that is for any tile anywhere from zero to nine moves, and apply these to a copy of the grid. Secondly, iterate through all the tiles in the second row and apply as many moves to said tile as the value of the tile above in the top row - for any other amount of moves, the tile above would never achieve the value zero. Continue the process of applying moves in a similar manner for all rows. Lastly, check if all the tiles in the bottom row are zero, in which case the puzzle is solved.