If this is the case, the equation either needs a domain readjustment(so, valid from t∈[0.7,∞]) or an equation redefinition.

Using a piecewise function would look like: 
$$
M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0.7,∞]
$$
$$
M \frac{dv}{dt} = 0, t∈(0,0.7)
$$

Friction, is just a reactionary force. For rolling resistance can be modelled as an example of a dynamic friction. [Hence, until motion occurs][1], $$F_t = β_1$$
Then, you should get better looking profiles.

Another way to do this would be to have a criteria, where the reactionary $β_1$ is just as big as $F_t$.

Therefore, 
$$
M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0,∞]
$$
Where
$$
\beta_1 =F_t, F_t∈[0,\beta_{max}]
$$
$$
\beta_1 =\beta_{max}, F_t∈[\beta_{max},∞]
$$

Here, $\beta_{max} $ is your initial value of $\beta_1$

  [1]: https://www.researchgate.net/figure/Static-and-dynamic-friction-behaviour-70_fig3_326623707