When do you replace an iphone 5s battery
Your answer can be found here: Battery Service and Recycling.

iPhone Owners
Your battery is designed to retain up to 80% of its original capacity at 500 complete charge cycles.
The one-year warranty includes service coverage for a defective battery. If it is out of warranty, Apple offers a battery service for $79, plus $6.95 shipping, subject to local tax.

My personal experience is that replacing the battery at about 700-800 cycles is a must. But obviously it depends on how the phone is used.
And yea, I'd say the cycle count is kinda irrelevant. Cycle count gives you a good idea of what state it is in... But when you are the one who uses the device, you should know when you've had enough.
I've always used those 3rd party phone repair shops, where the battery replacements tend to be relatively cheap.