The main difficulty in proving Baker's theorem is in estimating $C$. If you don't care about those estimates, then the proof is not difficult. For example, Chapter 7 of Waldschmidt's *Diophantine approximation on linear groups* gives a simple proof for the homogeneous case $\beta_0=0$. Some technicalities (like using Fel'dman integer-valued polynomials) are not necessary if you don't care about strong estimates.

The homogeneous case is slightly simpler because one can work with just $G_m^n$ or (dually) with $G_a^n$. However, the basic structure of the proof is the same:

- Using group structure, consider several points (working with $G_m^n$, the points are $(\alpha_1^s,\dots,\alpha_n^s)$).
- construct an auxiliary function/polynomial either using Dirichlet principle (in the form of Siegel's lemma) or using interpolating determinant
- Show that this polynomial with some derivatives is small (using either Schwarz's lemma or Hermite's formula) at those points.
- Show that it can't vanish at all those points.
- Show that if the polynomial or its derivative is small at one of those points, the value must be zero.
Assumptions, that $\beta_i$ and $\alpha_i$ are integers simplify the last step - instead of Liouville's inequalities, you are just using directly that if $x$ is an integer with $|x|<1$ then $x=0$. However, the other steps are the same.

Baker's theorem can also be proved using the Schnieder-Lang criterion, but I don’t know if that method will provide estimates sufficient for your needs.