Homothetic preferences from indirect utility
Your conclusion is a bit hasty:
(i) why did you implicitly exclude the case $a^i(\textbf{p})=0$?
(ii) if your individual is rational then $v^i$ is homogeneous of degree zero in $(\textbf{p},w^i)$ which has implications on $a^i$ and $b^i$... Your demand system should also satisfy the budget constraint (adding-up).