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In my original post, I used the very first brush I created, which was 125 pixels. This is larger than my immediate need, but — just as an experiment —  I enlarged it beyond 125 pixels and then ran into my pixelation issues.

After one of your posts, I went back and created a second custom brush, starting from a vector shape at 2500 pixels. With that new huge brush, there are no pixelation issues at any (smaller) size.

Maybe I'm wrong, but I always thought that the standard set of Photoshop brushes were somehow immune to pixelation issues because of a command or technique that I don't know about. But perhaps it's actually the case that all photoshop brushes were originally created at the maximum 2500 x 2500 pixel size, so it's just an illusion that they have some kind of special non-pixelating quality.

Here's the vector shape, if you're interested.