I think there is one more point worth making here.  There is a suggestion that
dimensional analysis can underpin a units system.  Actually, the idea that all
units can be broken down into a small set of fundamental units is very limiting
and results in many vexing issues.

For example, consider currencies.  There are currently hundreds of national
currencies and thousands of cryptocurrencies.  They all have the same basic
fundamental unit of “value”, but value is only loosely defined.  Furthermore,
there is no fixed ratio between the currency and its value.  It varies over
time, over location, and from person to person.

Consider units of a particular commodity.  The example of a ream of paper was
recently mentioned. A ream is 500 sheets of paper.  However two reams may not be
comparable.  They may refer to a different size of paper or a different quality
of paper.  So all prices for reams of paper would have the same fundamental
units of “value per each”, but both “value” and “each” are not necessarily
comparable.  In effect, the fundamental unit system is not complete.  You also
need to include information about what you are measuring.  For example, “each”
could represent a single item of anything.  The unit is not complete until you
include a description of what that anything is, and in effect, there is an
unlimited number of things it could be.

Now consider the issue of “unitless units”.  In electrical circuit we often talk
about gain, which it the ratio between the signal level at the output of
a circuit relative to the signal level at the input.  But you need to be
specific about how you measure signal level.  Typically, it would be a voltage,
current, or power level.  Consider a circuit where both the input and output are
measured in volts.  Then the gain would have units of "V/V", which is unitless
in dimensional analysis.  But units of "V/V" (voltage gain) is much different
from units of "A/A" (current gain) or "W/W" (power gain), even though they have
the same dimensions.  Mixing them up results in errors and confusion.  An
additional complication is that sometimes logarithmic units are used.  For
example, decibels in voltage, or dBV, is 20*log(Vout/Vin).  Again,
a dimensionless quantity, but nonetheless "dBV" much different from "V/V".

The same issue occurs with the arguments to trigonometric functions like sin(),
cos() and tan().  Generally, we assume the arguments are given in radians, which
is a dimensionless number.  But it could also be given in degrees, another
dimensionless number.  Radians and degrees are indistinguishable from the
perspective on dimensional analysis, but mixing them up results in errors.