iPhone Heating/Slow/Quick battery drain after repair
Nothing at all in your description contravenes the Apple diagnosis of swapping the battery. Part of battery service is that they can inspect for other damage and re-quote you if you need something other than a battery.

You didn’t mention if your device was stable enough to run the battery diagnosis from Apple Support. If that ran, I’d be even more sure they know what they are talking about and the chance they’re wrong grows less.

Your symptom of battery charge level varying more than 10% in any 300 second period is very clearly a faulty circuit or faulty battery. There really aren’t other options. The diagnostic will show full charge capacity, so you can calculate the max current draw divided by the capacity if you’re skeptical Apple’s diagnosis makes sense.

Let’s talk about water instead of charge for a moment. If my math is correct a FINA competition pool is 2,500 m^3 so let’s assume your battery has exactly that much mAH. Figure the size of the pipe to drain the pool when it’s new. Now a few years later, it can’t go from 80% full to 20% full in 5 minutes. If you measure that sudden drop in water level, the pool either has a massive leak or your measuring device is massively off. There’s no other option that’s reasonable.

The circuitry defines the measurement in iPhone battery, the hardware defines the max charge and max draw. The battery either works as expected or has faulty chemistry / circuitry and behaves out of bounds. That’s the main reason why a battery that’s not used up gets asked to be swapped and also why the diagnostics can point to the correct next step.