I ran "rustup component add rust-docs", and found that the HTML files actually were already installed on my macOS system under ~/.rustup.
I tested my VS Code installation first with the rust-analyzer extension installed, then with rls extension installed.
For my Rust hello-world test source, the parameter-hint pop-ups appeared only with the rust-analyzer extension installed.
That suggests to me that maybe the problem is in rust-analyzer extension code, in which case it might be helpful for me to file a bug report.
When I click the link for "format!" in the parameter-hint pop-up for a println! in my test code, there is an unsuccessful attempt to open the file "/std/macro.format.html".
As a test, I made a symlink so that /std/macro.format.html could be opened. Clicking the link in the pop-up actually did open this file in text mode in VS Code. I'm guessing that the intent is to instead open this HTML file in a web-browser tab.