Calculating the structural integrity of a pixel grid
So I found a solution using maximum flow algorithm with some tradeoffs.
The setup involves:

Connecting the source node to every node in the grid with an edge capacity equal to the nodes mass.
Connecting the sink to the bottom layer
Connecting every node in the grid to each other with some capacity defined by the node (i.e. the maximum stress/load from that direction)

After Dinic Max Flow algorithm runs, I traverse the residual graph starting at nodes which do not get the maximum flow from the source (i.e. nodes that can't apply their full mass) and find all nodes which have an edge with max capacity.
[CodePen Example]
Example configuration:

So the algorithm works much faster than mine, and that is because it does not spread mass evenly. As shown above the white squares are nodes which have edges with maximum flow. This does not necessarily mean that the node is overloaded, and as you can see there are alternative flows that are not at full capacity.
However since my goal was to simply find the overloaded nodes and destroy them (and floating parts) this does not matter so much.