The little bird that detects male channeling is obviously one of the ornaments that Rand did not recognize. Particularly because Demandred didn't recognize it either.

As to what a FULL paralis net is - maybe it is a ter'angreal like Mat's ORIGINAL medallion, which unravels weaves FULLY, as opposed to a partial one which would be similar to Elayne's weak copies of Mat's medallion, which can only unravel weak weaves.

However, thats where Cyndane's unfamiliarity with weave dissolving ter'angreal would be a problem.

In truth, there is only one logical conclusion, that fits all the quotes.

The paralis net itself existed during the War of Power. Lews Therin wore the first male version. However, the paralis net itself:

Does NOT detect male channeling and does NOT unravel weaves. Those two functions are fulfilled by some of Cadsuane's additional ornaments, which Lews Therin could not identify.

The paralis net therefore does something ELSE, probably associated with Cadsuane's well, since Lews Therin specifically associated it with Cadsuane being able to channel in Far Madding.

This is therefore the solution. Saidin detection and weave unravelling are not functions of the paralis net. They are functions of some of Cadsuane's other ornaments, which did not exist in the Age of Legends.