is the logit transform ever actually computed in modeling process of logistic regression?
Basically the Logit function is a series of transformations from ordinary regression. Transformations from ordinary regression are often called "link functions" It goes:

Ordinary Least Squares (OLS) -> Logistic Function -> Logit Function

You probably already know these equations but I am going to write them out anyway:



OLS gives you a single predicted value

OLS = β_1 X_1+β_2 X_2+⋯β_i X_i



The Logistic Function gives you the probability that Y = 1

Logistic Function = 1/(1+exp⁡[-1(OLS)])



The Logit Function gives you the log odds that Y = 1.

Logit Function = log_e(Logistic Function / (1-Logistic Function))



You are asking whether or not the logit function is ever actually used and the answer is yes, but you probably don't see it because a model summary will display the coefficients (betas) but not the equation that they're in. Your preferred language will certainly use these equations when estimating your coefficients, but there is no reason to show them to the user because the user is the one who specified the link function in the first place.

glm(y ~ x1 + x2, data = my.df, family = binomial(link = "logit"))