You do not want to kick variables out of the model based on preliminary T-tests since doing so may introduce an omitted variable bias. This is why the common practice is to estimate the full model first than use t-tests of the coefficient estimates in combination with adjusted R-squared, AIC, BIC, or other criterion to reduce the model.
Recall that retaining the null in your T-test does not mean the true correlation coefficient is 0 for sure, it means that there is insufficient evidence to conclude otherwise.
If it does turn out that the true values of the insignificant coefficients are 0, including the insignificant coefficients in your model will increase standard errors. However, if the true values of the insignificant coefficients are non-zero, even slightly, omitting them will make every coefficient estimate in your regression model bias.
Generally we would rather have a slightly higher standard error knowing the coefficient estimates are unbiased then have biased estimates with low standard error.
To see why a reduced model would be biased consider a simple example where the true population model is
y=\beta_0 + x_1\beta_1 + x_2\beta_2 + \varepsilon
Where $\beta_2 \neq 0 $ but perhaps very close to zero. If we omit $x_2$ on the bases of a prior t-test then our new model becomes
y=\beta_0 + x_1\beta_1 + w
Where $w= x_2\beta_2 + \varepsilon $. If $x_1$ and $x_2$ are correlated (almost always the case) then the correlation between the error term $w$ and $x_1$ is non-zero.
This is a problem because the least squares model assumes/ forces the error term to be uncorrelated with the independent variables, so estimating the reduced model via Least squares results in incorrect (biased) estimates of all the coefficients.
I could go into more depth about the mathematical reasons for why this true but I figured you saw that already. The thread Regression coefficients significance and threads cited therein provide more for that.