It still absolutely amazes me when people don't know what's in their tap water, or understand how to treat it, or how their tap water may differ from say, my tap water.
Water companies will generally use one of two treatments for tap water, chlorine or chloramines. The difference between the two is that chloramines consist of both chlorine and ammonia.
Why ammonia? Compared to chlorine, chloramines are more potent and much more stable. They don't "gas off." If you have low levels of chlorine in your tap water, by the time it hits your fish tank, it can practically be non existent.
What happens when you treat chloramine laced water with a basic dechlorinator? Look at the molecule above, the bond between the atoms gets broken and the chlorine is neutralized, but you're left with ammonia. This is why chloramine dechlorinators like Prime must also neutralize ammonia, they need to in order to treat chloramines fully.
Try coming over to my house and putting your fish in my heavily chloramine laced tap water without a conditioner and see what happens!
The reason Seachem recommends dosing for the entire tank volume with Prime is that levels of chlorine or chloramine are not consistent in most places. A couple times a year here we have algae outbreaks in our water collection point and the water gets a considerably bigger dose of chloramine. It's cheap insurance. It's not going to hurt your fish. Prime can be safely dosed at 5 times the normal dose in emergencies, and probably more than that.
In the end, take 10 minutes and look up what's in your tap water from your water company's website. Call them if you have any questions, they staff people whose sole job is to monitor water quality, and IME they are quite happy to chat about it (most people probably find their job boring and don't care to hear about it). You may be able to get away with sodium thiosulfate...
Screw it, googled it and here is some light reading:
http://www.skepticalaquarist.com/chlorine-chloramine
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