Time zones have nothing to do with jet lag; jet lag would still exist without them, if you had jets. Time zones were created initially by railroad companies to coordinate train schedules. Previously you'd have each station set its "local time" by a definition of noon, or their own company standard, so it could vary. (Getting rid of time zones would just mean that you'd return to a "local time" definition, which you'd still be hours off of from jet lag if you crossed enough distance in a short enough amount of time.)

And yes, they really did matter: if you get "local time" wrong when trying to coordinate sharing of track, then you can end up with a train wreck. This did actually happen:

A critical period of change in how time was kept occurred in August 12, 1853. Two trains were headed toward each other on a single track. The conductor of one train thought there was enough time to switch tracks before the approaching train was scheduled to pass through. But the conductor's watch was too slow. The train sped around a corner and collided head on with the other train killing fourteen passengers. As a result of the accident, railroads ordered more reliable watches for their conductors and issued stricter rules for running on time and New England railroads agreed to set their clocks and watches to a single standard time. This was how standard time was set.

For a very nice discussion of time-coordination problems and their relation to other issues at the time, Galison's Einstein's Clock, Poincaré's Maps: Empires of Time is about how these sorts of questions factored into the work of a particular young patent clerk in Switzerland, who had numerous patent applications coming across his date regarding the coordination of clocks through electric means. It includes a nice discussion of the emergence of time zone standards (for which my above take is just the initial impetus — it gets more complicated very quickly).