One simple way to proceed would be to do a binomial logistic regression to determine statistical significance and confidence intervals, and then convert the results from the log-odds scale of the logistic regression to probability scales if you wish to present results in terms of probability differences or risk ratios.

With the standard logit link function in binomial regression, you model the log-odds of the event (in your case, a long_wait) as a function of predictors. You main predictor is Season, in your example a 2-level categorical variable for which we can take "Summer" as the reference level. Then the intercept of the logistic regression is the log-odds of a  long_wait in Summer, and the regression coefficient for Season would be the difference in log-odds between Winter and Summer; equivalently, this Season coefficient is the log of the odds ratio for a long_wait between Winter and Summer. If that regression coefficient is significantly different from 0 then the null hypothesis of no difference between Seasons is not supported.

You would take the individuals into account with a random effect in a mixed logistic regression model. If the values available for each patient are already aggregated among the 5 years of data collection then you would be limited to a random effect for the intercept of the logistic regression model, allowing individuals to differ in the log-odds of a long_wait in Summer while modeling a single log-odds-ratio in the Season coefficient.

Many people (including me) have difficulty in thinking about odds and odds ratios, but those values are easily transformed into probability values, differences in probabilities, or risk ratios to make your presentation easier for your audience to understand.

If you use the R glmer() function in the lme4 package for this analysis, you would re-organize your data into a long form, with one row representing data for a single individual and a single season, labeled by individual and season in columns. The corresponding visit observations could be represented by two columns, one for the number of long_waits ("successes" if you model the probability of a long_wait) and one for the number of short_waits; alternatively, you could provide the proportion of long_waits as the outcome and use the total number of visits as a weight.

One simple way to proceed would be to do a binomial logistic regression to determine statistical significance and confidence intervals, and then convert the results from the log-odds scale of the logistic regression to probability scales if you wish to present results in terms of probability differences or risk ratios.

With the standard logit link function in binomial regression, you model the log-odds of the event (in your case, a DNA) as a function of predictors. You main predictor is part-of-week, in your example a 2-level categorical variable for which we can take "weekday" as the reference level. Then the intercept of the logistic regression is the log-odds of aDNA on a weekday, and the regression coefficient for part-of-week would be the difference in log-odds between weekend and weekday; equivalently, this regression coefficient is the log of the odds ratio for a DNA between weekend and weekday. If that regression coefficient is significantly different from 0 then the null hypothesis of no difference between weekend and weekday is not supported.

You would take the individuals into account with a random effect in a mixed logistic regression model. If the values available for each patient are already aggregated among the 5 years of data collection then you would be limited to a random effect for the intercept of the logistic regression model, allowing individuals to differ in the log-odds of a DNA on a weekday while modeling a single log-odds-ratio in the part-of-week coefficient.

Many people (including me) have difficulty in thinking about odds and odds ratios, but those values are easily transformed into probability values, differences in probabilities, or risk ratios to make your presentation easier for your audience to understand.

If you use the R glmer() function in the lme4 package for this analysis, you would re-organize your data into a long form, with one row representing data for a single individual and a single part-of-week, labeled by individual and part-of-week in columns. The corresponding appointment observations could be represented by two columns, one for the number of DNAs ("successes" if you model the probability of a DNA) and one for the number of non-DNAs; alternatively, you could provide the proportion of DNAs as the outcome and use the total number of appointments as a weight.