Three steps. About a minute. Because Swift Client isn't signed by Apple yet, you'll need to clear macOS's quarantine flag with one Terminal command — that's the whole trick.
Double-click Swift-Client-mac.dmg in your Downloads folder. A small window pops up showing the Swift Client app icon next to an Applications shortcut.
Drag the Swift Client icon onto the Applications folder shortcut. macOS copies it across — when the bar fills, the install is on disk. You can eject the .dmg now.
Open Terminal (Spotlight → "Terminal"), paste the command below, and hit return. It strips the "downloaded from the internet" attribute Apple set on the app, so Gatekeeper will let it open.
$xattr -cr "/Applications/Swift Client.app"
Launch it from Applications or Spotlight. On first launch you'll sign in with your Microsoft account, then pick a Minecraft version and hit Quick Play. The launcher takes it from there.
Apple charges $99/yr for a Developer ID, and Swift Client is a free hobby project — so we haven't paid for one yet. Without a signature, macOS marks downloaded apps as quarantined and refuses to launch them. The xattr -cr command clears that flag on Swift Client only. The app itself is unchanged and the source is on GitHub if you want to audit before running it.