> But I'm not sure if this answers my question. What exactly is this command doing? I'm guessing it sets "black" and "rich" to 1 and test if that minus the base case = 0.

It does, it your model (model type, predictor terms) is correctly specified. 

It depends on the coding of variables `black` and `rich`. If `black = 1` for a black person and `black = 0` for a white person and `rich = 1` for a rich person and `rich = 0` for a poor person, then the sum of coefficients of `black` and `rich` indeed measures the difference in probit (inverse probability function of a normal distribution, or normal quantile value, or linear predictor) of `mort` between a black rich person (`black = rich = 1`) and a white poor person (`black = rich = 0`). For example, if $\beta_\text{black} = 0.5$ and $\beta_\text{rich} = -0.7$, then $\beta_\text{black} + \beta_\text{rich} = -0.2$ which means a black rich person has a lower predicted risk of mortality than a white poor person, controlling for all other predictors in the model. 

Nevertheless, risk is usually measured and best understood as a probability instead of quantile (i.e., probit or logit). Because normal and logistic cumulative probability curves are nonlinear, the effect of any predictor on probability is nonlinear and depends on all predictors. A `-0.2` difference in probit can be between quantiles at 1 and 0.8, or -0.4 and -0.6, or something else, which correspond to probabilities at .8413 and .7881 (-.0532), or .3446 and .2743 (-.0703 difference). So we should translate the contrast in linear predictor to probability to facilitate communication. See tutorials at https://marginaleffects.com/chapters/categorical.html. 

However, you have a few other highly correlated variables `hisp`, `other`, and `middle_class` which may not be treated constant when varying `black` and `rich`. Can I person still be in the middle class without being rich? It is useful to show a percentage breakout of different combinations of these categorical groups. Also, when categorical predictors are involved, it is very important to examine interaction effects: add terms like `black:rich` `black:age` `college:rich`. The combined effect of being a black AND rich person may be different than the sum of the effects of being black and being rich separately. See my answer to https://stats.stackexchange.com/questions/639430/frank-harrells-interpretation-of-interaction-in-regression-results/639444#639444. 

> What is the correct way of doing this?

Mortality and disease risks are usually studied in survival analysis instead of binary regression. It seems your data report the age of a person at death. A relevant model will be

    survival::coxph(Surv(age, mort) ~ black + rich + black:rich + ...)

This assumes that `age` is continuous (i.e. calculated as days instead of years). If the data only record integer age, discrete survival analysis will be more appropriate. See discussions at https://stats.stackexchange.com/questions/57191/ and book Tutz, G., & Schmid, M. (2016). Modeling discrete time-to-event data. Springer Nature. https://doi.org/10.1007/978-3-319-28158-2. It essentially entails `glm(mort ~ s(age) + black + rich + black:rich + ...), family = binomial("cloglog"))` where `s()` is a smoothing curve and the data must be restructured so that each patient has several rows, each for a year until the death year where `age` increases by 1 for each year. For your data, probably `age` is the only predictor that varies by row within the same patient. 

Note that both `glht()` and `marginaleffects::comparisons()` conduct Wald tests, which require a well-behaving likelihood function (sufficiently approximating to a quadratic curve at its maximum). Basically it means the coefficient and its standard error should not be very large. For the Wald test caveats, see https://stats.stackexchange.com/questions/193643/likelihood-ratio-vs-score-vs-wald-test-different-p-values-which-to-use/637205#637205. To conduct likelihood-ratio on linear combinations of coefficients, which many test routines do not provide directly, see https://stats.stackexchange.com/questions/648266.