> When the response variable is a proportion (example values include 0.23, 0.11, 0.78, 0.98), a quasibinomial model will run in R but a binomial model will not.

A binomial model *can* be estimated with proportions as outcome. See below:

    prop.m1 <- glm(cbind(Successes, Total) ~ X1,
                data = df,
                family = binomial)

    prop.m2 <- glm(Proportion ~ X1,
                 data = df,
                 family = binomial,
                 weights = Total) # provide prior weights