That is correct. The subscription service is a rental, like all media subscriptions services.
Spotify, Amazon Music,. YouTube etc... all work the same way. You get access to the songs and albums while you are subscribed, but you do not own the songs from it. So they stop being available when you cancel it.
If you want to own the songs then you need to actually buy them from the iTunes Store.
Like a Gym subscription. You don't own the gym equipment, you don't get to take it home, you only get to use it while you pay for the subscription within the confines of the gym.
Otherwise, what's to stop anyone from downloading 10,000 songs for $9.99 for one month, cancelling it and then selling them out for profit. Piracy is something the media rights owners are very sensitive about and would not let Apple or anybody else for that matter do something like that.