[Quoth]

You download the books OUTSIDE of Calibre and import them!

HTML is a less good format to download from Gutenberg. Better to download epub or mobi (called Kindle now) and then convert it if you want HTML. But HTML on its own is poor for ebooks. That's why Mobi and epub were invented as they are really a zipped directory with HTML files for the body, typically one per chapter, CSS (typically two files), a system index, a resource file listing what is in it, image files and font files, if used.

HTML is not a sensible format for any sort of proper ereader. The HTML is only for people that want to use a web browser, which is historic and madness today.

Don't implement a reader that simply uses Gutenberg on demand. That's "cloud madness philosophy". Have a browser that can download, or a directory to import to and have the reader only use local files.

Calibre is primarily a program to manage ebooks already copied to the computer. It imports a copy. Then you can manage metadata, searches and conversions and transfers to an ereader, or storage on phone/tablet that has an ereader app.
Oddly there is a full contents search for epubs as an option in "Quality Check" tool.

Normally epub is the best format to use. But Mobi is older. Some mobi may have old mobi and Kindle KF8/azw3 in the same file! However I find downloading "Kindle Format" from Gutenberg works best. Then I import that to Calibre, convert to ePub2 (using various options to fix quotes, remove paragraph space and have 1.4em first line indent, and embed Georgia font. Set line height and minimum line height both zero to allow user to change it, subset fonts etc).

Then I check the cover and other metadata in Edit Meta data. There are plug-ins to search very many websites.

I make sure the author name is consistent and correct before a metadata search.

I was using Gutenberg when Kindle format was called Mobi, epub didn't exist, Kindle didn't exist and the first eink (by Sony) was later that year.
Palm PDAs and Symbian Phones were probably the first gadgets other than laptops you could read ebooks on before dedicated ereaders existed. Gutenberg is THAT old, which is why text and HTML are offered.