There is nothing stopping you from fitting the second model you described, if the theory or prior empirical research supports it. Your model covariates are limited only by your imagination and how you want to interpret the parameter estimates. However, your second model is not a "substitute" or "alternative" to multivariate modeling, as the interpretation of the parameters differ between the two. Consider the models you described above:
$PctBodyFat_i=\beta_0+\beta_1Height_i+\beta_2WaistSize_i+\epsilon_i$ (1)
and
$PctBodyFat_i=\delta_0+\delta_1WaistHeightRatio_i+\nu_i$ (2)
In Model 1, $\beta_1$ is interpreted as the expected change in $PctBodyFat$ corresponding to a one unit increase in $Height$ after controlling for $WaistSize$. Meaning, among people of the same $WaistSize$, what is the average (expected) change we see in $PctBodyFat$ when $Height$ increases by one unit. Similarly, $\beta_2$ is the expected or average change in $PctBodyFat$ corresponding to an increase in $WaistSize$, all other things equal. That is what "controlling for" means, holding all other values constant and changing only one parameter at a time. In Model 2, on the other hand, the interpretation of $\delta_1$ is, the expected change in $PctBodyFat$ corresponding to a one unit increase in waist-to-height ratio. Do you see how these two models are not equivalent?
Here's a worked example based on what you described above using Body Fat Data. I used Stata in this example. I also converted height to cm so everything would have the same unit of measurement.
. *model 1 multivariate
. reg pctbodyfat heightcm waistsize
Source | SS df MS Number of obs = 252
-------------+------------------------------ F( 2, 249) = 274.24
Model | 12090.3059 2 6045.15297 Prob > F = 0.0000
Residual | 5488.68364 249 22.0429062 R-squared = 0.6878
-------------+------------------------------ Adj R-squared = 0.6853
Total | 17578.9896 251 70.035815 Root MSE = 4.695
------------------------------------------------------------------------------
pctbodyfat | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
heightcm | -.1458789 .0319761 -4.56 0.000 -.208857 -.0829009
waistsize | .6423569 .027589 23.28 0.000 .5880194 .6966944
_cons | -14.31075 6.042646 -2.37 0.019 -26.21196 -2.409532
------------------------------------------------------------------------------
$Height$ and $WaistSize$ are both statistically significant at p<0.001 in this example, with a 1 cm increase in $Height$ expected to decrease $PctBodyFat$ whilst a 1 cm increase in $WaistSize$ expected to increase $PctBodyFat$.
So far so good. Now here's the output from Model 2 where a waist to height ratio was entered instead of height and weight separately:
. *model 2 ratio
. reg pctbodyfat wthratio
Source | SS df MS Number of obs = 252
-------------+------------------------------ F( 1, 250) = 214.53
Model | 8118.31801 1 8118.31801 Prob > F = 0.0000
Residual | 9460.67157 250 37.8426863 R-squared = 0.4618
-------------+------------------------------ Adj R-squared = 0.4597
Total | 17578.9896 251 70.035815 Root MSE = 6.1516
------------------------------------------------------------------------------
pctbodyfat | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
wthratio | 69.829 4.767533 14.65 0.000 60.43935 79.21864
_cons | -17.28263 2.517475 -6.87 0.000 -22.24079 -12.32447
------------------------------------------------------------------------------
Again, waist-to-height ratio is highly statistically significant in Model 2. The reason the coefficient is so huge is because waist-to-height ratio is less than 1 for most people, so a 1-unit increase in waist-to-height ratio can be expected to produce a fairly large magnitude in change for $PctBodyFat$. If you're uncomfortable with the large coefficient, you can multiply waist-to-height ratio by 100 and your model coefficient will be divided by 100 (i.e., 0.69829), but it's a matter of preference and what makes sense. What is interesting to note here is that Model 1 appears to have a better fit than Model 2 based on the adjusted R-squared values.
After fitting a regression model, we can predict outcome values for observations based on certain characteristics. Suppose we want to know what the predicted values are from each of the 2 models for people with these characteristics.
$$
\begin{array}{phwws}
Person & Height & WaistSize & WaistHeightRatio \\
\hline
A & 170 & 80 & .47058824 \\
B & 152 & 80 & .52631579 \\
\end{array}
$$
From model 1, we expect the following percentage body fat for the 2 hypothetical cases above:
. margins, at(heightcm==170 waistsize==80)
Adjusted predictions Number of obs = 252
Model VCE : OLS
Expression : Linear prediction, predict()
at : heightcm = 170
waistsize = 80
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 12.27839 .509845 24.08 0.000 11.27423 13.28254
------------------------------------------------------------------------------
. margins, at(heightcm=152 waistsize=80)
Adjusted predictions Number of obs = 252
Model VCE : OLS
Expression : Linear prediction, predict()
at : heightcm = 152
waistsize = 80
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_cons | 14.90421 .9258657 16.10 0.000 13.08068 16.72773
------------------------------------------------------------------------------
We expect Person A to have a percent body fat of 12.3 and person B to have a percent body fat of 14.9. Now if we were to predict percent body fat for the same exact people using Model 2, we would get:
. margins, at(wthratio=(.47058824 .52631579))
Adjusted predictions Number of obs = 252
Model VCE : OLS
Expression : Linear prediction, predict()
1._at : wthratio = .4705882
2._at : wthratio = .5263158
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 15.57807 .4578963 34.02 0.000 14.67625 16.4799
2 | 19.46947 .3881273 50.16 0.000 18.70506 20.23389
------------------------------------------------------------------------------
Using Model 2, both persons are expected to have higher percentage body fat than in Model 1, 15.6% and 19.5% for Person A and Person B, respectively. To summarize
$$
\begin{array}{phwws}
Person & Height & WaistSize & WaistHeightRatio & E(PBF|Model_1) &E(PBF|Model_2)\\
\hline
A & 170 & 80 & .47058824 &12.3 &15.6 \\
B & 152 & 80 & .52631579 &14.9 &19.5 \\
\end{array}
$$
Of course, as I stated above, there is no stopping you from entering your variables in the model every which way you like. It all depends on the theory, your preference, and what you want your model parameters to mean as far as your research question is concerned. If you have the relationships between the variables worked out beforehand and want to enter it that way, then great, but combining variables before entering into the model is still not an alternative to multivariate modeling.
As you say, this is a simple example. In reality, there is probably no getting around multivariate modeling. In fact, you may even lament the fact that you do not have enough variables to control for in your dataset, and you start worrying about endogeneity and other sources of bias in your model.
