I have a dataset (unfortunately cannot disclose any part of it) which has a binary response variable. For each independent variable, I calculate the log odds of the positive cases given each value of the IV and plot them to check linearity, i.e., x-axis is the IV and y-axis is the $logodds(DV=1|IV)$. I find that at least 50% of my variables (which includes interaction effects) are not linear in log odds with the DV. How does my model converge then? Is the linearity assumption not a very strong one or any model can converge but in such cases simply cannot be trusted? If any additional information is required, just let me know and I will try to my best to provide to clarify my question even further.
As a side question, I am wondering if my approach of plotting logodds against each IV to check linearity is correct because everywhere else, people just advice to use the Box-Tidwell approach.