In the linear probability model (probit or logit), should $\sum_i \hat p_i = 1$?
No, it would only be a strange coincidence if the predicted probabilities summed to $1$. Consider a case where there were $N=100$ data with $90$ 'successes' and $10$ failures and the null hypothesis was true for all predictor variables. Then the true probability of 'success' for each point would be $\pi = .9$ and the sum would be $100\times .9 = 90 \ne 1$. This would hold for either a logit or a probit model.
In the real world, probabilities will rarely sum to $1$. More specifically, with an intercept only model you would need to have only $1$ 'success' ($\hat p = \frac 1 N \Rightarrow N \times \frac 1 N = 1$).