Hi AdahS,
To answer your question, yes, you can test the battery first thing; as a matter of fact, you can just plug in a new battery before you even remove the old in order to test it. All you have to do is remove the rear cover, disconnect the fingerprint sensor, then take out the wireless charging coil. At that point you have access to the battery connector and you could unplug the old battery and plug in the new one. I can't say for sure, but chances are good the phone will boot without the fingerprint sensor or the charging coil attached; if it does so successfully you'll be able to see the amount of charge on the battery and whether plugging it in correctly charges the replacement battery.
If it works with the new battery, you can then shut everything down, unplug the new battery and proceed on with the battery replacement procedure.
On the other hand, if plugging in the new battery doesn't help anything, then yeah, it may well be time to replace the charging port. For testing, I would reassemble it up to the point of putting the charging coil back in and see if it boots without the back cover (and thus the fingerprint sensor). You might have to temporarily plug in the fingerprint sensor if it refuses to boot or let you log in without it.
But before you start on that path, let me check a couple of things with you. First of all, please tell me you've tried a different charger and a different cord, right? I'd feel pretty stupid if I went to all the trouble to replace a charging port only to find out I had a bad cord!
Secondly, I'd think it would be worth the investment to pick up an inexpensive wireless charger in order to try and isolate the fault to either the charging port or the battery. Since the wireless charging uses an entirely different set of components and circuitry, it would point pretty definitively to either the port or the battery depending on whether it charges wirelessly or not. You can pick one up on eBay for under $5 USD, so to me it would be a worthwhile investment.
One final note; if you decide to go ahead with the charging port replacement, as long as you're in there anyway I'd suggest replacing the battery as well whether it needs it or not. Your phone is five years old now, and that's getting close to the lifetime for the battery. You're right there in the neighborhood anyway and it's only a few extra steps to take the old one out.