Update : I brought my phone to the local authorised service provider, and they did a factory reset for me.
Whatever they did, it worked. My phone came back to me with every faculty working as good as new (I did several factory resets on my phone previously, everything was wiped out, but problem remained).
Then, I did a restore from my backup back at home (this was partly an experiment to further this thread, in pursuit of the cause). The problem (bluetooth searching, wifi grayed out) returned!
I went back to the service provider for another dose of "factory reset" - this time, the symptoms remained.
Since my phone was under warranty (luckily), I was given a new one (service provider claims it's not refurbished, it's completely new). I can only surmised those without a warranty would have to pay.
So in conclusion :
1) The ios 7.0.3 contains a bug that could cause the phenomenon.
2) The afflicted phone could be repaired with a factory reset (this was done at the authorised service provider who could have used a version that came with the fix I am not sure)
3) The bug can be replicated by restoring the backup.
Aside to Apple : Please do the right thing. People who have suffered or are suffering from this fiasco should at the very minimal get a FOC replacement.