Electrical – What could cause the bathroom exhaust fan to stop working

You need to identify the line, the load, and each of the switched wires. When everything is disconnected and the power is turned on, the line is the wire that will be hot. Use a non-contact tester to locate this, and follow the appropriate safety precautions to avoid electrocuting yourself (if you don't know these, it may be time to hire an electrician). The line could be coming directly into the fan box, in which case it's likely the separate pair of two wires. Or it may be coming in at the switch.

Next, turn on each switch and see which wires become hot, again with a non-contact voltage tester. Note which wire belongs to which switch. If there's no current going to the switch, then the line isn't connected anymore and you can instead use a continuity test from the wires in the ceiling with each switch turned on individually. This can/should be done with the power turned off.

Finally, determine where your load is going, if you have any. This is going to be any remaining wires, and you'll know where they go by all the items that don't work when the power is switched on. If the pair of wires in the ceiling isn't the line, then it's the load.

My best guess, updated from your video's findings, but without being there to test anything or get a better look at how wires are tied together in the back of the switch is:

• The line is connected at the switch.
• From the switch, it is likely running a non-switched hot to the fan box to connect to the load inside the bundle of 6. There will also be a neutral running in that bundle of 6.
• The other wires in the bundle of 6 are each individually switched hots. You'll need to test each one and label them. They should also be marked with a piece of black tape on each at the fan so there isn't any mistaking them for a neutral.
• The 2 wires in the ceiling are likely the load to another location.

And to hook it up, if my guess is correct:

• At the fan you'd wire the hot non-switched hot to the load. Pigtail the neutral from the bundle of 6, the neutral going to the load, and the neutral going to the fan. Then the switched wires are connected to the fan depending on each task.
• If your new fan doesn't have the extra lights (and therefore doesn't use the extra switched wires), then you should disconnect them on each end, bind and cap all unused wires together, and label them as unused with an indication of where the other end of the wire is. I'd also replace the switch with one that only has the necessary number of switches (otherwise everyone will be trying to figure out what the unused switches do).
• Since it sounds like your fan has a ground, and I don't see one in the picture, I'd strongly encourage you to add one. The conduit itself may be grounded (you'd detect continuity from neutral to the conduit in that case). But modern wiring practices, along with the fan you purchased, expect a dedicated ground wire.

Your test shows you have hot coming from the breaker (hot to ground 120 volts) but you are loosing it on the neutral side. I would start at the breaker box again only this time looking at (volt meter test) the neutral wire. If it's good go to the first junction box and test there and so on.