To show the silliness of the proposed idea, let's consider getting rid of all our basic units of measure and just go with one. Length. The meter, the yard, the light-year, it doesn't really matter. Everything can be measured with our 1-unit system and school kids will thank us forever.
Let me show you how to do it. For our given unit length - call it the Spevy, denoted by ჷ . Light in a vacuum will span that distance in a certain time interval t. t = ჷ/c. If you take c as dimensionless, t is now measured in length as well. And now we are off to the races. Literally. Velocity is next. That is a length over time, both of which have identical units and so velocity is dimensionless. Similarly, acceleration would have units of inverse length.
The gravitational force between 2 masses is proportional to the mass and inversely proportional to their separation, so that is the yet-to-be defined mass squared over length square, F = M2/L2. This is also equal to mass times acceleration F = M a. Set these two equal and solve for M: M = a L2. Since acceleration has inverse length, mass has a unit of length.
No more moles kids! It's all Spevy's from here on out. What's a Spevy? It's something about that long. Doesn't that make chemistry so much easier? It's all just distance. No more worrying about counting the yotta-number of atoms sitting around in a flask, just pull out a Spevy-stick and go at it.
Equally extreme would be to make every measurement as a basis unit with no relationship between any one measurement and the other. It would be the Humpty Dumpty approach where the units means just what I choose them to mean — neither more nor less. 
The best approach is to find a middle ground, somewhere between 1 and infinity. The SI system is at 7. But even at that, there are thousands of other units that people still use because they are convenient. Astronomers use light-years not to cause confusion, but to reduce it by keeping the numbers manageable, just as particle colliders measure areas in barns (and millibarns and...). And other units are valuable because they can be more clearly understood. Take gasoline mileage. In the US, we use miles per gallon, but this can easily be reduced to an inverse area. Miles are a length, and gallons are a volume, a length cubed. Divide a length by a length cubed and you end up with an inverse length squared = inverse area. Elsewhere in the world, liters per 100 kilometers is used for mileage, so that can be reduced to an area. But in either case, is any information more clearly communicated than using the non-standard units?
As the letter-writer states, the mole arose during a previous time when our knowledge of atoms was so much less, but that doesn't mean that the unit should be discarded anymore than we should discard other units developed in earlier times, such as the second, minute and hour (all of which were based on the presumption that astronomical motion was so much more precise than it actually is, hence the need for leap years and leap-seconds.)
I'm not worried that anyone is going to stop using the mole at any point now or in the future. We have too much existing literature that would be unintelligible if that change could suddenly occur. And while I very strongly emphasize for the difficulty of learning high school chemistry , there are far more challenging ideas to be learned as the education continues. If someone decided not to pursue a chemistry career because they didn't understand the mole, they were not going to have a successful career in chemistry regardless of whether the idea was taught or not. Quantum mechanics, despite it's probabilistic nature, would guarantee the end.
 That's a backwards 'S' to further emphasize how backwards this whole idea is. The HTML code is & #4343 ; if anyone wants to pick up on the idea and run with it.
 When you look at the yields of some published reactions, you sometimes get the feeling that that reality is closer than we think.
 I know this from personal experience. I had a truly horrible high school teacher that left me ill-prepared for college chemistry. By the second week of freshman chem, I was already in trouble. But I put in the extra time and succeeded.
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