I fixed the issues by reflowing the entire main board.
I did this in the oven at 190 degrees Celsius for 10 minutes. Before putting it in, I removed all the rubber and plastic parts from the board. I created feet for the board to elevate it off the tray (using four balls of aluminum foil on the corners and one in the middle to prevent warping). I also applied flux to all the larger chips. I suspect that the issue was with IC 3000, a processor that supplies the drive-on signal, but I thought it would be worthwhile to apply flux to everything.
I reconnected everything, and to my surprise, it worked!
I'm not sure how long this fix will last, but at least it's functional for now.
By the way, I realized that the part number of the board is EAX65608606. I can find a replacement AU version for $300 AUD, but I found a CA version for only $38 AUD on Taobao (the Chinese domestic market version of AliExpress). If the TV breaks again, I might try the CA board. From what I can see, there's an extra antenna port on the CA version, which I'm not sure what it's for (perhaps for cable TV, which might be different in Australia).