After reading "On the Rule of the Road" by A.G Gardiner, what do you think would happen if all of us began to assert our liberty?

We can see what would happen in such a situation simply by looking at what Gardiner says in the first paragraph of “On the Rule of the Road.”  There, Gardiner says that

the end of such liberty would be universal chaos. ... Individual liberty would have become social anarchy.

In other words, our society would be uncontrollable if we all took advantage of our liberties.
To understand this, just think about the two examples that Gardiner gives.  What if there were no traffic rules?  This would be terrible.  People would drive their cars in whatever way they wanted.  There would be so many traffic accidents as people tried to go through intersections at the same time or as they tried to drive very fast in congested areas.  Pedestrians would never be able to cross from one side of the street to the other as drivers would never stop for them.  This would be chaos.
Imagine a similar thing in a store or at a movie theater.  Imagine that people did not get into line to pay for their items or to buy a movie ticket.  We would have chaos here, too, as everyone tried to push to the front to get the cashier’s attention.  Older people or children would never be able to manage.  Fights would surely break out.  Again, there would be no order whatsoever.
Gardiner’s second example has to do with courtesy to others.  What if people were free to make as much noise as they wanted whenever and wherever they pleased?  My neighbor could play loud music in the night while I was trying to sleep.  I could retaliate by honking my car horn incessantly outside his house as his baby was trying to nap.  Similar things could happen if people did not exercise any courtesy while walking along the sidewalk.  People would crash into one another as each tried to exercise their liberty to walk down the middle of the sidewalk.
In this essay, Gardiner is trying to show us how we can only have a decent society if we curtail our own liberties.  If we insist on doing whatever we please, we have a chaotic society where we actually lose our ability to do anything.  By giving up some of our liberty, we get a livable society in which we are able to do almost everything we wish in some way.