I haven't saved spinach seeds, but I expect most seed stalks to look "very dead" before the seeds are fully ripe and dry.
The longer you can leave them "on the vine", the more likely they are to reach full maturity and viability.
Reasons to harvest earlier:
1. You expect constant rain that would rot the seeds before collecting.
2. The seeds are about to drop spontaneously and be lost. You can bag some stalks to catch the seeds, or just watch very closely and as soon as any seeds drop, harvest the brownest 2/3 of your plants. If you don;t have rain or heavy dew, set out boxes or trays or whatever to catch the first seeds to drop. Once you see any seeds in your trap, harvest several stalks or shake the stems - FULLY ripe and dry seeds will fall right out of some plants with just a shaking.
3. Birds or neighbors are likely to steal them before you can harvest them.
4. Gardener impatience. There's no cure for that, other than (perhaps) saving lots and lots and LOTS of seeds over many years. My theory is that, eventually, the obsession will fade or at least become more manageable. I hope you let me know if that ever happens for you!
