My issues with Wi-Fi also started after updating to 6.1.3. I've got iPhone 5 and it's unbearable. No matter where I am, whether the network is protected or not, the connectivity drops every 10-30 seconds, sometimes stays on for a little longer, then reconnects again or not. After a couple of connects and drops it usually can't see the network anymore until I move some switches back and forth.
There are wifid crash reports in the Diagnostics panel. I had no problems with Wi-Fi at all before the update, it was working like a charm, and now it's pretty much useless. I've noticed that turning on/off Bluetooth does help a little with restoring Wi-Fi connection after it drops. Moving BT switch seems to force a deamon restart or smth, so it comes back (only to drop again soon after).
And yea, had the wifi and bt switch greyed out too when I restarted the phone when the the drops occur. It was really scary, but the switch became active after an hour or so. I've no idea whether its hardware problem or not, since it really didn't happen at all before 6.1.3 for me. It doesn't look like it.