Diagnostics said my battery was as good as new. But my experience with the phone tells me it runs down noticeably more quickly than when it was new. So who is right?
Did they give you an exact number? The industry standard is 80% original battery capacity. If it's 3 years old I would expect that it would at the very least be down to at least 90% if it's been used a lot. I certainly wouldn't expect new. If you have a Mac, you can try using coconutBattery. There's also a free app called "Battery Life". I think it's kind of clunky and it sends the user to a lot of ads, but it's free and it does show basic battery health reduction.
A lot of issues that seem like they're battery related are more about usage patterns or perhaps iOS changes that affect power consumption. In order to get a longer battery life, the power management has to be able to reduce power consumption, but without noticeably affecting the user experience.
I wouldn't necessarily say it isn't worthwhile to get a new battery. As far we know this battery promotion ceases at the end of 2018, and certainly it's not a bad thing to have a completely new battery. Apple's previous policy was that they or an AASP wouldn't perform a battery service unless it didn't pass the standard battery diagnostic for diminished capacity.