How can I get an older version of Final Cut Pro on macOS Mojave?

I have tested this in a Mac running macOS 11 Big Sur, fully expecting it to work, but unfortunately I confirm your findings. It used to ask "do you want to get the most recent compatible version" (or something to that effect), but now it just says that you need macOS 14.6 or later.

Assuming you have clicked on the iCloud icon in your purchase history to download FCP, that message usually suggests Apple is not keeping an older version available for download, or at least not old enough for your version of macOS.

Not much to do there unfortunately.

If you are desperate to use FCP the only real solution is to spend more money on a new computer.

I can recommend the base model M4 mini which is heavily discounted (£530 in the UK) and which will blow your mind.

You will probably find it is in the region of 10 times faster together with countless other improvements . . . money well spent.

Correct, Apple has actually limited severely the past versions available recently.

The only "problem" with the base model is the 256GB SSD but for another £60 you can get a 1TB USB-C NVMe (SSD) which gives you a total of 1.25GB for less than the cost of the basic M4 from the Apple Store.