Possible alternative:
Add a small / decreasing value to dist inside munch_data()

This does the trick and has near-zero performance hit. Coefficient "2" was chosen arbitrarily, based solely on the looks of the result.

extra <- pmax(floor((dist + (1 - exp(-dist * 2))) / segment_length), 1)

I've tried to tackle this issue, but I don't think my geometry is enough to solve this properly. A more general solution should rely on curvature (rapid change over small distance = more subsegments). Perhaps there is a ready-made algorithm for this.

Maybe its wildly wrong, but its my best attempt. Seems to fix an issue for lines near origin in polar coordinates, and doesn't seem to explode with subdivisons elsewhere:

# Calculate the change in angle of the tangent vector
d_angle <- abs(diff(atan2(diff(data$x), diff(data$y))))
d_angle <- c(d_angle, d_angle[1])

extra <- pmax(ceiling(dist * d_angle / segment_length), 1)

Maybe segment_length could be exposed to end-user somehow?
I would gladly use it, because now my solution for these types of charts is to create subdivions myself before plotting or to monkey-patch ggplot2.

I think it probably might be more straightforward to just 'lie' about at what radius point is in the distance calculation.
The code below is for demonstration purposes only and not a recommendation for how to write ggproto classes, but it shows what I mean by lying. We can get a smoother finish at near-minimum radius points by instead of rescaling to [0, 1], we simply rescale to [0.2, 1] so that the distance calculation thinks the radius is larger.

library(ggplot2)

df <- data.frame(
  x = 1:3,
  y = c(2, 4, 20)
)

cp <- coord_polar()
new <- ggproto(
  NULL, cp,
  # similar to CoordPolar$distance()
  distance = function(self, x, y, details) {
    if (self$theta == "x") {
      r <- scales::rescale(y, from = details$r.range, to = c(0.2, 1)) # changed
      theta <- ggplot2:::theta_rescale_no_clip(self, x, details)
    } else {
      r <- scales::rescale(x, from = details$r.range, to = c(0.2, 1)) # changed
      theta <- ggplot2:::theta_rescale_no_clip(self, x, details)
    }
    ggplot2:::dist_polar(r, theta)
  }
)


ggplot(df, aes(x, y, fill = factor(y))) +
  geom_col(width = 1, show.legend = FALSE) +
  scale_y_continuous(limits = c(0, 50)) +
  new +
  theme_void()

^{Created on 2023-07-26 with reprex v2.0.2}

Huh, its great at its simplicity! Thanks!

But is it something that would be added into default coord_polar() or would I have to re-implement it for my usage?

I have no idea about what adverse effects this might have as I'm not too familiar with calculating polar distances, but visual tests don't seem too much affected. Until I better understand that calculation, I think I'll hold off on putting in a PR. You're of course free to take this experimental adjustment for your own usage.