I am trying to understand whether there is a statistical foundation for a formulation of a multiple regression problem as a combination of multiple simple regressions. Say we have a response variable value $y_1$ (a single data point of $y$ for the sake of example) and 3 independent variables with observed values $x_{11}$, $x_{12}$, $x_{13}$ for that sample. In a standard multiple regression, we aim to find $w_1$, $w_2$, $w_3$ triple that minimizes below error:
$(y_1-x_{11}w_1-x_{12}w_2-x_{13}w_3)^2$
And here is a formulation of the same problem a little differently, such that it minimizes below error instead:
$(y_1-x_{11}w_1)^2 + (y_1-x_{12}w_2)^2+(y_1-x_{13}w_3)^2$
The latter formulation is essentially the sum of the errors from 3 one-variable regression problems considering one variable at a time in a simple linear regression.
I am wondering whether there is a method which formulates a multi-variable regression problem using the latter error formula. If not, why is that, i.e., what is wrong with that latter formulation?