Animate Probability Density Function with gganimate

Question

I am trying to replicate the beta distribution animation of Wikipedia in R.

Originally, I started with the following code

ggplot() +
  xlim(0, 1) +
  geom_function(fun = dbeta, args = list(shape1 = 5, shape2 = 5))

But I did not manage to use gganimate to animate the plot based on the shape parameters.

Therefore I tried a second approach where I first generate the data and then plot it using geom_line() (similar as in this post)

library(tidyverse)
library(gganimate)

# get beta PDF for specific parameters ------------------------------------
calc_beta <- function(alpha, beta){
  X <- seq(0.01, 0.99, 0.01)
  PDF <- dbeta(X, shape = alpha, shape2 = beta)
  tibble(X, PDF)
}

# alpha and beta values ---------------------------------------------------
n <- 50
r <- exp(seq(log(0.1), log(5), length.out = n))
alpha <- c(rev(r), r, rep(5, 2*n), rev(r), r)
beta <- c(rep(5, 2*n), rev(r), r, rev(r), r)

# tibble with alpha, beta and PDF -----------------------------------------
dat <- tibble(alpha = alpha, beta = beta) %>% 
  mutate(time = 1:n()) %>% 
  group_by(time) %>%
  mutate(plotdata = map2(alpha, beta, calc_beta)) %>% 
  unnest(plotdata)

# plot --------------------------------------------------------------------
p <- dat %>% 
  ggplot(aes(x = X, y = PDF)) +
  geom_line() +
  transition_time(time) +
  coord_cartesian(ylim = c(0, 4)) +
  annotate('text', x = 0.5, y = 4,
           label = 'alpha = {round(alpha[as.integer(frame_time)], 2)}
          beta = {round(beta[as.integer(frame_time)], 2)}') +
  labs(title = 'alpha = {round(alpha[as.integer(frame_time)], 2)}
          beta = {round(beta[as.integer(frame_time)], 2)}')
animate(p, nframes = 400, fps = 40)

Now I have 3 Qustions

1. Is there a way to use gganimate to animate the args of geom_function (first approach)?
2. Is there an easier solution then the second approach?
3. Why does the animate labeling not work within the annotate() function but only within the labs() function?