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 we know
the scale of our problem, we can use _fixed point_ arithmetic with respect
to this scale to solve that problem.

Sometimes, this is a valid approach, and this is the one picked 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 either 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 to 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, so that loss of precision caused my multiplications in fixed point arithmetic do not occur.

Most of the times, it is however a terrible approach, here are a few
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 a system to
   appropriately represent all constants you need
   might not even be possible. Think to quantum physicits working
   with infinitely small particles whose speed is near the speed of light.

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 the
   solar system setting you described, we can represent Pi with 6 decimal places,
   which gives a terrible accuracy when computing the circumference of
   a planet orbit, for instance.

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

   Assume that we still do not know the value of the gravitational constant. We would
   make a lot of measures and write a computer program to find an approximation of that
   constant.  Unfortunately, in the solar system setting you described, the gravitational 
   constant is represented by 0, which should be the, rather useless, result of our     
   computation.

4. Some mathematical functions will not work well with fixed precision
   arithmetic, because of their growth rate.

   The most important ones are the exponential and the gamma function, which practically
   means that every program working with anything else than polynomials will be flawed.

5. In fixed point arithmetic, it is very hard to multiply and divide
   numbers correctly.

   This is because if we do not know _a priori_ the size of the numbers, we cannot tell
   if their product will fit in the representation. That is, we would have to check
   manually for precision underflow before each multiplication.

# Conclusion

While the conclusion of your question implies 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 which 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.