Why does my beta regression look linear when plotted? Im using a beta regression to model since my dependent variable is confined between [0,1]. However, when I plot the relationship between x and y via my beta regression, the effect looks linear. When doing a linear scatterplot, the lines are almost identical. This is unlike any other beta regressions that I've seen so far, and I wonder what I might be doing wrong.
My code looks like this:
library(betareg)

vb=betareg(dep ~ pctDFNB, data=v, link="loglog")
vb2=betareg(dep ~ pctDFNB, data=v, link="logit")

# PLOT
ggplot(v, aes(x = pctDFNB, y = dep)) +
  geom_point(size = 4, aes(fill = party), shape = 21) +
  scale_fill_grey() +
  geom_line(aes(y = predict(vb, v),
                colour = "log-log", linetype = "log-log")) +
  geom_line(aes(y = predict(vb2, v), 
                colour = "logit", linetype = "logit")) +
  scale_colour_manual("", values = c("red", "blue")) +
  scale_linetype_manual("", values = c("solid", "dashed")) +
  theme_bw()

And the plot I receive looks like this:

For information, my beta distribution looks like this: 
EDIT: Thanks for your answers, they are truly insightful and appreciated!

As the other answers have already noted, your relationship is almost linear, and this is picked up by the beta regression. (More precisely, by the logit link. This is where any "curved" shape would come from, not from the shape of the beta distribution as such. You may be misunderstanding the roles of the logit link function and the distributional assumptions.)
Compare a beta regression model for the built-in ReadingSkills dataset:

We see how the relationship between iq and accuracy is almost linear, as in your data, for participants with dyslexia - but it is noticeably curved for participants without dyslexia.
R code:
library(betareg)
data(ReadingSkills)
iq_predict <- with(ReadingSkills,seq(min(iq),max(iq),by=.01))   # for plotting predictions

with(ReadingSkills,plot(iq,accuracy,pch=19,las=1))
with(subset(ReadingSkills,dyslexia=="yes"),points(iq,accuracy,pch=19,col="red"))
with(subset(ReadingSkills,dyslexia=="no"),points(iq,accuracy,pch=19,col="blue"))
model <- betareg(accuracy~iq*dyslexia,data=ReadingSkills,link="logit")
lines(iq_predict,predict(model,data.frame(iq=iq_predict,dyslexia="no")),col="blue")
lines(iq_predict,predict(model,data.frame(iq=iq_predict,dyslexia="yes")),col="red")
legend("bottomright",lwd=1,pch=19,col=c("blue","red"),legend=c("No dyslexia","dyslexia"))

