logit ScalingFunctions

Having learned how to make symlog axes in this question, I was interested in the logit scale, the transforms of which are

logit = {
Function[x, Log[x/(1 - x)]],
Function[y, Exp[y]/(1 + Exp[y])]
};


These functions are inverses of one another, as required. Then, trying

Plot[x, {x, 10^-3, 1 - 10^-3}, ScalingFunctions -> logit, Frame -> True]


I get a figure

There are two problems. First, I get a message in the Messages window,

 Power::infy: Infinite expression 1/0 encountered.


which causes the red frame. Second, only the first set of ticks above 0.5 are there (up to 0.9, further decades are missing)

Weirdly, when I evaluate lots of other expressions immediately after plotting---0, for example---the red frame goes away.

I think the issue may be an imprecise evaluation but using SetPrecision as suggested in that question does not seem to fix the issue.

Questions:

• Where are the top half of the ticks? How do I get them to show up?
• How do I prevent the red frame and divide-by-0 error?
• a work-around: ticks = Quiet@ Append[{logit[[1]]@.9, "0.900", {0.03, 0.}, AbsoluteThickness[.1]}]@ ChartingScaledTicks[logit, "TicksLength" -> {.03, .02}][ logit[[1]][10^-3], logit[[1]][1 - 10^-3]]; Plot[logit[[1]]@x, {x, 10^-3, 1 - 10^-3}, Frame -> True, FrameTicks -> {{ticks, ticks}, {Automatic, Automatic}}]`
– kglr
Aug 11 '21 at 22:32
• The failing of the ticks is confirmed as a bug by Wolfram support. Aug 16 '21 at 21:22
• The error message + red frame was fixed in 13.0 but the missing ticks were not. Dec 15 '21 at 11:54