Why do downloaded applications arrive as .dmg files

Since applications are just copied to the drive anywhere (/Applications is simply a nice place for multi-user systems but apps run anywhere they land as long as it's readable) there is no mandatory place to record the installation.

Have you considered using Time Machine to construct a history?

I use BackupLoupe which will index all of your backups, so you can search and know the first backup that has that App. It's not a true installation time, but it's close within a week at worst if your backup runs at least that often.

You'll have to add some software that explicitly inventories your mac periodically to assemble such a list. Most of these apps are geared to remote management and not cost effective for one or a handful of macs.

Technically, a .pkg is not a file, but a folder. So maybe collecting it into a single file is essential for downloading.