Why do I need to subscribe to Apple music to have music I PAID for on my devices?

You don't have to pay to have your music on all your devices.

Music purchased from the iTunes store will appear on all devices signed in to the Apple ID used to purchase it automatically.

For Music obtained for somewhere else, payment is only required if you want to use iCloud Music Library, which is part of 2 subscription services. It is not a stand alone service.

If you want to the convenience of iCloud Music library then you need to subscribe to either Apple Music or iTunes Match which have a cost associated with them, since you are getting more than just the iCloud Music library.

Otherwise, you can manually sync all your devices to the computer that holds the music library to to get the library on each for free. See here: Use the Finder to sync your iPhone, iPad, or iPod with your computer - Apple Support

Purchased from where exactly?

If your Mac is running Catalina, simply select the Library on the side Pane in the Music App it should show all all your purchased music there.

The Browse and For You sections are specifically for the Apple Subscription Service. If you have one of those sections selected, it will always require a subscription. But it has nothing to do with your own music.

For your Music, make sure Artists, Albums or Songs under Library is selected, and your Apple ID is signed in under the Account Menu.

Again, the iCloud Music Library is part of two subscription services. You need to pay for either of them if you want to have music not bought in the iTunes Store in the iCloud Music Library.

Music you bought from the iTunes store does not require this, and is automatically available. You do not need to turn on iCloud music for those purchases.

If you need to use the iCloud Music library, then yes you need to pay for one of the subscription services, as they are not free.

Make sure all of your devices are signed in with the Apple ID used to buy that music from the iTunes Store.

Home Sharing works only when all devices are connected to the same Wifi network.

You will not have access to the music when you are away.

You can of course use Home Sharing to transfer the Music from an old computer to anew one and have everything on the new computer if that is what you want to. But you never mentioned an old computer.