How do you know if you need to add a sub (without auditioning one, I mean)?

To answer your question if you need a sub, get a sub with a Bluetooth App. SVS and Paradigm are two of the few offering that feature. With a swipe of your finger you can mute the sub. A/B ing with only a few tracks will let you see the NIGHT and DAY difference. The most important part is you are doing this from the listening position.  No more running behind the sub. A quality sub will fill in the missing frequencies 40-20 Hz or lower, even the most pricey floor standers can’t produce. Check out Paul’s of PS Audio YouTube videos on the subject. Even my Golden Ear Triton 2’s which have powered bass modules built in benefit from my SVS SB-3000!

Just TRY a sub. We're beating a dead horse now.

A video I watched said even a violin sounds better with a sub. That did it. 
I was associating a sub with that annoying thump from a car truck and didn’t understand why anyone would be happy adding that to a 2 channel system. I’m new to better hifi and quickly learned you don’t know what you don’t know. 
Just found out my second sub is on the way. Planning for all four by years end. 
Whatever you do, do it!

I use a pair of subs with my main stereo pile (RELs bought used in great shape...inexpensive...I now own 3 of them) and this does help with the dreaded standing wave issues, but I also experienced a huge improvement when I used just one sub. Note that the world in general, meaning simply the sound of life, has a very wide frequency range...most speakers can cover from maybe mid bass to relatively high frequencies, but subs provide a more life-like element including some ambient sound as well as accurate bass and often make your main speakers sound much better. I check where my main speakers drop off and put the subs close to that point...adjust for taste...I also use a Loki sometimes if a recording sounds like it needs help from Knob Turner, and I have a new version coming from Schiit (back ordered) and it will be interesting to see how they improved the thing.

@optimize mentioned some good points as have some others. To address your question: the sound will definitely be better with a sub or two or more. Even if you have a speaker that can truly reach down to 20Hz the position of your speakers will dictate how the room nodes develop. If you move the speakers to a position where the bass is good it is extremely unlikely that you will be happy with the mids and tops!

A speaker that produces from 40Hz and upwards with 3 subs optimally positioned will easily outperform  the necessarily huge and expensive full range speakers.
You can also rest assured that all domestic rooms will need some acoustic treatment which is another issue.

Along with some good advice I see the inevitable recommendation to add tone controls or room EQ. which will not sort out the unavoidable peaks and nulls. A 40Hz tone has a wavelength of about 28ft which when bouncing around will combine constructively (peaks) and destructively (nulls), using room correction the peaks can be attenuated so they are less offensive at the listening position but when you move around you get to hear all the peaks that remain. With nulls, which is musical information that is missing, understand that no device can manufacture that info, it's gone AWOL. Trying to pump lots of power into that frequency will just cancel with the same power. It's futile and if you drive the amp into clipping you stand a good chance of blowing your tweeters.

EQ can work once the Schroeder frequencies are taken care of which varies from room to room, about 300Hz average. How to take care of them? Add bass traps and multiple subs and as has been mentioned, get a mic.and free program off the net. If you go this route  you will be utterly amazed and will have little desire for any EQ!