If you turn the Switch off and plug in the charger does it show the charging indicator?
Might just be that it's not drawing the full 2.1A, so it's not able to actually charge while powered on.
Some chargers won't provide more than 500mA, because there was no real standard for fast charging until USB-PD showed up, and although fast charging with powerbanks and unofficial generic 2A+ wall chargers does usually work, that's only because they try to support as many different standards as possible, while the Switch won't support every standard. According to tests I saw using different wall adapters, most would provide up to 2A but some did only provide 500mA.
The way USB wall chargers used to fast charge was that they would sense the data pins (which aren't used on a charger, obviously), often that means sensing the resistance across the data pins, or it could be just sensing a short across the data pins, and based on that would either supply 500mA or supply up to the full amperage they can handle. These days it's more complicated and relies on actual communication between the charger and the device, this is also the way USB-PD works, but for older chargers that's not a thing.
I do have a USB power meter, but sadly, I don't have my Galaxy Note 3 charger to test with, I sold my Note 3 a while back. I believe that's the same charger used in the S4. I can't say I've ever really used it to fast charge other things, I only charged things that would only take 500mA anyway. But said USB power meters are only a few bucks on AliExpress. Usually they're called USB meters or USB testers, and they will show how much power is drawn.
