Why would you pay $20 for a transaction, if you can pay much less for it?
It is going to be encouraged by your greed.
Unfortunately this won't work this way.
Look at segwit. SegWit got introduced in July (i think). And still the amount of segwit transactions hovers around 10% (http://segwit.party/charts/#).
Because the most commonly used wallets don't really suport it yet, so most people don't have choice as they are not tech savvy and can't do command line.
Even people who made Segwit in the first place have not really released the segwit functionality for the wallet they make.
So what do you expect from others?
This is not unusual. If you ever worked in a big organisation, you know the release cycle (preceded by design, development and testing) of such a feature would take probably about a year.
The Lightning Networks are surely much more complex and will take far more time to release than the segwit payments - but it's gonna happen for sure, as there is money to save and therefore to make.