Alpha removal with ImageMagick is bad but only with a white background

I'm getting surprising (to me) variations in antialiasing quality when converting a PDF (with black text on white only) to a PNG and removing transparency with ImageMagick (version 6.7.7.10 from Ubuntu 14.04).

My sample input file is the result of running pdflatex on

\documentclass[preview]{standalone}
\begin{document}
Hello, world!
\end{document}

At 300 dpi, the result is horrible (that's with a white background, adding -background \#ffffff or -background white produces visually indistinguishable output):

convert -density 300 -alpha remove -alpha off a.pdf 300-white.png

There's some anti-aliasing, but not very good, and some lines (such as the horizontal stroked on the e) have completely disappeared.

Choosing any color other than white produces a decent result.

convert -density 300 -background \#fffffe -alpha remove -alpha off a.pdf 300-fffffe.png

Curiously, the rendering is better at low resolutions (while still not ideal) — at least the text is readable.

convert -density 100 -alpha remove -alpha off a.pdf 90-white.png

At higher resolutions, the rendering is still clearly poor, but at least the lines start to appear.

convert -density 750 -alpha remove -alpha off a.pdf a.png

• Why is antialiasing so bad for black on white, and perfectly fine with any other background color, even if that background color is visually indistinguishable from white?
• If I set this up as part of an automated process where someone won't be around to visually inspect the result, what parameters do I need to avoid to get good results?