Why doesn't time change in the non-relativistic limit of Lorentz transformations? A simple boost in the $x$ direction is given by:
$$ \Lambda = \begin{pmatrix}
\cosh(\rho) & \sinh(\rho) & 0 & 0 \\
\sinh(\rho) & \cosh(\rho) & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{pmatrix} $$
Which get linearized to the following transformation:
$$
x^0 \mapsto x^0
, \quad 
x^1 \mapsto x^1 + \frac vc x^0
$$
How come the zeroth component is not linearized to $x^0 \mapsto x^0 + \frac vc x^1$? Is that because there is another factor $c$ in the time components? Since $x^0 = ct$, that would mean the time is transformed like
$$ t \mapsto t + \frac v{c^2} x,$$
and $c^{-2}$ is just so small that is ignored?
Or is it just to fit the Galilei transformation?
 A: The Lorentz transformations :  \begin{pmatrix}
\cosh(\rho) & \sinh(\rho)  \\
\sinh(\rho) & \cosh(\rho)  \\
\end{pmatrix}
form a group.
The galilean transformations : 
 \begin{pmatrix}
1 & 0  \\
v/c & 1  \\
\end{pmatrix}
form a group
But the transformations : 
\begin{pmatrix}
1 & v/c  \\
v/c & 1  \\
\end{pmatrix}
do not form a group.
If you restrict your transformations to a group structure, which is the simplest hypothesis, you cannot keep the last example of transformations. 
A: The Lorentz boost has two different low-velocity limits: the Galilean transformation appropriate for transforming ultra-timelike four-vectors, which is usually what we're interested in if we want to recover low-velocity kinematics, and the whimsically named "Carroll transformation" appropriate for transforming ultra-spacelike four-vectors. Your proposed time transformation is part of the latter.
Intuitively, the effect of the Lorentz transformation is a rotation along curves of constant spacetime intervals:
 Image from here.
If we zoom in on the upper vertices of the hyperbola, the low-velocity Lorentz boosts of timelike vectors with $ct\gg|\mathbf{x}|$ are approximated by:
$$\begin{eqnarray*}ct' \approx ct\text{,}&\quad&\mathbf{x}' \approx \mathbf{x} - \mathbf{v}t\text{.}\end{eqnarray*}$$
Treating this low-velocity approximation as a transformation in its own right, we get the Galilean boost:
$$\begin{eqnarray*}ct' = ct\text{,}&\quad&\mathbf{x}' = \mathbf{x} - \mathbf{v}t\text{.}\end{eqnarray*}$$
This is sensible, because the tangent lines to the hyperbolas on ultra-timelike vectors are horizontal, so a low-velocity boost should not change the time coordinate by an appreciable amount.
If instead we zoom in on the right vertices, and repeat the above procedure on ultra-spacelike ($ct\ll|\mathbf{x}|$) vectors, we get the Carroll transformation:
$$\begin{eqnarray*}ct' = ct - (\mathbf{v}\cdot\mathbf{x})/c^2\text{,}&\quad&\mathbf{x}' = \mathbf{x}\text{.}\end{eqnarray*}$$
Note that this reflects the fact that a low-velocity boost of an ultra-spacelike vector  does not change the spatial coordinates appreciably, as the tangent lines to the hyperbolas are vertical there.
On a purely formal level, they are equally valid low-velocity limits. The Galilean transformation is much more physically significant because material objects should follow timelike vectors, not spacelike ones, so we can interpret it as a possible transformation between inertial frames formed by ideal clocks and rulers (or some other means). In contrast, the Carroll transformation does not allow a sensible interpretation as a transformation between physical inertial frames.

“My dear, here we must run as fast as we can, just to stay in place. And if you wish to go anywhere you must run twice as fast as that.” ― Lewis Carroll, Alice in Wonderland

