Steam

If it's not a game you can redeem through Steam, then install through desktop mode and add it to your list of games in Steam (add non-steam game). That will let you access it through the normal Steam OS

Lutris integrates with itch.io these days, I haven't used it extensively but it seems robust enough. It should also support your local DRM free games and whatnot.

GameHub supports Humble Bundle and itch.io though I haven't tried it on Steam Deck yet.

Edit: Unfortunately the appimage/flatpak packages are pretty borked and installing via pacman would be more trouble than it's worth.

Non-steamlaunchers can make it easier https://github.com/moraroy/NonSteamLaunchers-On-Steam-Deck