Just a thought but there are two versions of WiFi - 2.4 Gb and 5 Gb. In the UK 2.4 is the only one I've come across but it is possible that an IPhone acting as a Wi-Fi AP could use 5GB 802.11 a/h/j/n as this is a legal arrangement in the USA. The Extreme AFAIK will only use the 2.4Gb flavour 802.11b/g/n.
Do you know what Wi-Fi Channel your IPhone is on ? You should be able to find this using the laptop and INSSIDER software if the IPhone itself won't tell you. I'm very surprised that Wi-Fi analyser wouldn't see the Hotspot. Does it show nothing at all.
On the Apple support site I see the following :
Personal Hotspot requires a supporting hotspot tethering plan from your carrier. The maximum number of supported connections is five, but this number may vary depending on your carrier or iPhone model.
While devices are connected to Personal Hotspot, the status bar will be blue and will indicate the number of connected users.
Could the problem be that you are exceeding the number of simultaneous connections if other devices are connected as well.
The other thing to try would be to change the channel your hotspot uses in case you are being swamped by another AP on the same channel- shouldn't happen I know but worth a try anyway.
I also saw the following on the Apple site :
Note: If you leave the Personal Hotspot screen or put your iOS device to sleep, the device only broadcasts the Wi-Fi network for an additional 90 seconds. If no devices join the Wi-Fi network during that time, your iOS device will stop broadcasting your Wi-Fi network. You can start broadcasting your Wi-Fi network again by tapping Settings > Personal Hotspot.
Could this be what is happening ?
BobC