I do something similar to @lpb. I use the desktop app, so immediate browser search is unavailable. So from a task I generate the task-link weblink, and then manually open that in the browser. So text search is a huge pain.
The Asana model appears to be that descriptions are short, so who needs search. But for my usage, in some cases it is convenient to put my task documentation directly into the description field. I’ll put the “description” portion at the top, and then use headers or what not to organize the rest of my content. So like the OP, I’m putting actual content in the description, and want to be able to search over it.