Usually if I suspect a cooling failure I would also consider removing the heat-sinks to check the ICs underneath, however the pcstats article mentions that the heat-sinks are attached to the memory ICs with "frag tape", which usually means the thermally conductive tape that can be difficult to remove without accidentally ripping one of the components off the board.
While there are no visible damage on the capacitors, capacitors can fail without showing any visible damage, so replacing them in an attempt to make the card working again is a possibility, note that the Geforce 4 MX440 was released in 2002, so it might have lead-free solder, so I would recommend setting the soldering station to 20-30 Celsius above the usual. Pre-heating the board can be also considered seeing that it might already have heat damage, so pre-heating it to ease soldering is unlikely to cause any more damage than what already have occurred if the fan was replaced because the original failed.
"A process cannot be understood by stopping it. Understanding must move with the flow of the process, must join it and flow with it." - Dune