Your argumentation against floating point numbers is very fragile,
probably because of naivety. (No offense here, I find your question is
actually very interesting, I hope my answer will also be.)

> A classic argument is that floats provide a greater range, but high
> precision integers can meet this challenge now. For example: with
> modern 64-bit processors, we can do fast integer calculations up to
> 2^64. The solar system is a little less than 10 billion km in
> width. 10 billion km divided by 2^64 is about 5 microns. Isn't being
> able to represent position within the solar system to the precision
> of half a human hair enough?

You seem to do an implicit statement, according to which once I know
the scale of my problem I can use fixed point arithmetic with respect
to this scale to solve my problem.

Sometimes, this is a valid approach, and this is the one choosed by
Knuth to implement distance computations in TeX.  What makes the use
of fixed point arithmetic pertinent in this case is that all
quantities appearing within a computation are integers or distances
occuring in a typesetting problem.  Because the field of applications
is so narrow, it makes sense to choose a very small unit length—much
smaller than what the human eye can perceive—and convert all
quantities into multiples of this unit.  A very important remark, is
that in the typograhical problems relying on this representation of
numbers, we never need to multiply two lengthes together.

Most of the times, it is however a terrible approach, here are the
reasons why:

1. There exists physical constants and you cannot always adapt their
   units in a sensible way.

   Consider your solar system setting. The gravitational constant is
   6.67×10−11 N·(m/kg)2, the speed of light is 3.00x10+5 m/s, the mass
   of the Sun is 1.9891×10+30 kg and the mass of the Earth is
   5.97219×10+24.  In your fixed point setting, you will not be able
   to represent the gravitational constant to a satisfying precision.
   So you will change the unit.  But by doing so, you have to replace
   each number—replacing well-known, familiar quantities, by cryptic
   values.  Furthermore, it is very likely that finding an appropriate
   system is not even possible (think to quantum physicits working
   with the lightspeed fast infinitely small particles).

2. There exists mathematical unitless constants.

   The value of Pi is 3.1415 without any unit attached. There is
   actually a lot of similar useful constants that cannot be
   accurately represented in an arbitrary fixed point system.  In your
   solar system setting, you can represent Pi with 6 decimal places,
   which give a terrible accuracy when computing the circumference of
   an orbit, for instance.

3. In a fixed point system, you need to know in advance the size of
   the quantity you are computing.

   Say you do not know the value of the gravitational constant. You
   make a lot of measures, plug everything in the computer and… tada!
   You get a zero!

4. Some mathematical functions will not work well with fixed precision
   arithmetic, because of their growth.  Basically the exponential and
   the gamma function, for the most important ones.

5. In fixed point arithmetic, it is very hard to multiply and divide
   numbers correctly, because if you do not know the size of the numbers,
   you cannot tell if their product is representable or not—leading to
   large numerical errors.

# Conclusion

While your conclusion imply that fixed point arithmetic could be
sufficient for all-purpose computations and that floating point
arithmetic should be reserved to supercalculators, it is precisely the
converse that is true: floating point arithmetic is a very good and
very sensible tool for all-purpose computations, while fixed point
will only do well in very specific, well analysed, cases.