Pull the power board and replace all the electrolytic capacitors 20uF and above. Switched mode main power section isn’t reliably starting up. I’ve had this fix all (10+) but one. It’s so common there are sites selling kits by make & model.
I’ll wager the power outage was preceded by a surge which damaged the power supply. The standby switched mode section won’t reliability start when plugged in. Skilled help necessary to either fix or replace the board.
Completely common for a vacuum powered fuel pump on riding mowers. Follow the gas line from the tank and look for a square ~1.5” “sandwich” with 4 bolts and 3 hoses. Hoses are gas in & out and vacuum pulses coming from the crankcase. Disconnect the hose at the carb and crank the engine - no gas means pump diaphragm has failed.
The flashing is an error code, check the LG site for what. Power supply startup failures are “usually” capacitor failures in the switched mode power supply circuitry. You can see about a replacement or even fixing it. There’s at least 1 website that assembles kits of the recommended capacitors based on make & model.
A harness is for consistent easy assembly at the factory. You can direct wire with a connector or not. The ZTR 429 I’m rebuilding wouldn’t engage the blade clutch, the connector was so badly corroded. I bypassed it using wire nuts. Now waiting on new drive cones to finish the rebuild.
If yours is like mine, under the seat is a large (~12” x ~8”) plastic cover, held on by 4 bolts. Upper 2 are easy, lower 2 aren’t. Inside are the power and reversing relays. There’s probably is a fuse inline with the reversing relay coil. The cart I work on has a big, manual switch not a relay. I’d look for fuses and listen for the relay engaging before pulling any covers etc.