This is mistaken. Estimation of nonlinear models does entail solving first order conditions, numerically, rather than analytically, as can be done for linear models. With perfect collinearity, there is more than one solution to the first order conditions, and the matrix of second derivatives is singular. Most iterative gradient-based optimization methods (e.g., BFGS) will not work. It may be the case that certain procedures can still produce an estimate (e.g., regularization methods, as noted in another response, or using a derivative free optimization method) but the estimate will not be consistent, in general.
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