Showing it with Distance instead of Time solves most of the problems you mentioned.
100 units = 1 Teemo. To have a better notion of distance, Ezreal's Q is just above 1000 untis. Obviously it would have to be measured in thousands (with the 1K notation), but it shouldn't get too complicated, as Summoner's Rift is only about 20K in length.
Obviously, units is not something that can be used to estimate with precision the ETA of the pinger and I'll agree with yu there. However precision is not necessary. Anyone can easily acertain that if the pinger is 3k Units away, he should be there rather quickly, 1K he's probably just outside your screen, 10K should take quite a while. You don't need to know at which second your ally will arrive, but this system would give you a pretty decent idea.
Of course nobody would get that notion in their first few pings, but it would be a valuable skill that is very easy to learn.
Implementing it is extremely easy, as when someone pings "On My Way" the shorter path is already calculated and shown on the mini map. To calculate it, the system has to know the distance of the path. Implementing it is simply a question of displaying the number at the bottom of the ping animation.
As a side note looking at the mini map still won't give you precise information, as the shown path does not take into account blast cones, possible detours, or encounters with the enemy jungler. Most of the problems you brought up are already here. This system is in no way a tool to solve all of them (I'd even argue that some of them should not be solved for gameplay's sake), but it would be rather a Quality of Life change.
The new ping system is already a QoL change over the old ping system, and most of the arguments you made about lack of precise information can be made over changing from the old pings to the new ones, or even about having pings at all, given that you could give much more precise information over the chat.
By the end of the day, pings are about fast communication with a decent-ish degree of information given. In this case, upgrading the OMW ping is astonishingly simple with no drawbacks, as no new problems will be created, and a couple will be solved.
I personally don't see a single con to it, only pros.