From what I can tell, this occurrence sounds like something causes the flight controller to reset after a second or so, triggering it to reconnect with the ESCs as it initializes itself again.
The "du-du-du beep-beep" noises you point out are generated by the ESC (through the movement of the motors) to signify that they're receiving control inputs from the FC and are ready to go. More specifically, I believe that the "du-du-du" sounds (with ascending pitch) signal that the ESC is receiving signal pulses from the FC and the "beep-beep" sounds mean that the ESC is done with its own initialization process and ready to fly. (some ESCs also use the number of these pulses to indicate the number of cells in the battery they think they're connected to)
If the ESCs are detecting that the FC connects and then reconnects to them after a second or so, this likely means that something on the FC caused it to hard reset and reinitialize itself. This could be caused by anything from broken solder connections on the board to a voltage regulator that finally kicked the bucket. (or has begun to)
With respect to troubleshooting the issue and verifying that the issue is with the FC and not something else, diagnoses aren't very clear cut because the symptoms can vary from undetectable to glaring. Some dead giveaways are:
- Something on the FC is getting abnormally hot
- There are solder connections on the FC board which are visibly broken/intermittent
- As you say, a particularly hard crash can kill/degrade FCs
- Something just looks off with the FC during startup
As @Kralc helpfully mentions: (slightly paraphrased from original comment)
it can help to try unplugging parts to see if the problem changes or goes away. (Remember, it might also be a combination of parts that cause the issue, so it is good to keep notes of what you've tried) If the symptoms remain even with nothing else connected, it is more likely to be the FC.
Other symptoms are far more subtle and might not be reliably detectible by the eye. If you are able, I'd recommend trying to replace the FC if none of the above symptoms are present.