The brief running of the inducer suggests that the control is unhappy with one of its inputs (safety switches). Turn off the power to the furnace and with an ohm meter/continuity checker test the following one at a time by disconnecting either one of the leads and then measuring across the two terminals of the device:
- burner high limit switch (circled in green, mounted between the burner ports and the bottom of the collector box)
- flame roll-out switches (circled in red, one at the left and one at the right of the burner manifold)
- blower compartment high limit switch (circled in blue, mounted above the control board)
The two high limit switches are auto-resetting, so if the furnace is cool and either of those is open (infinite resistance) then it has failed and needs to be replaced. The flame roll-out switches are manual-resetting; if either of those is open then reset it by clicking the little button between the terminals and test again. None of these switches should trip in normal operation, but particularly if a flame roll-out switch trips, somebody needs to figure out why it's happening.
There should also be a pressure switch. It's not visible in any of your photos but I think the two yellow wires going to the upper-right corner of the photo lead to it. Check whether that switch is open or closed. I'd expect it to be open. If it isn't, then try disconnecting one lead and powering up the furnace. If this leads to the inducer running longer than the half-second or so shown in your video, immediately reconnect the lead to the pressure switch and observe what happens next. (This would be a sign that the pressure switch has failed closed.)
As an aside, there's a lot of corrosion at the bottom of the collector box cover and a fair bit at the inducer output too. Things are getting wet that shouldn't be wet. The corrosion at the inducer output would have to be caused by water coming down the flue; the stuff at the collector box cover could be from that or possibly condensate from the air conditioner, a humidifier problem, etc.
