I just like having 4 people in a raid battle. I tried thinking of ways they could incentivize people joining in other raids but it'd really just come boil down to "100% catch rate" (which you already have on your own Dens) "100% Hidden abilities or higher shiny chances" (which would thin the player base even more after everyone got what they wanted) and "more item rewards" (which if it were any way other than the same odds and items for all trainers then people wouldn't want to host raids either). Getting more people raiding by removing a $$$ barrier is the only way I saw a solution, and even then.. it's still temporary. I think filling more lobbies maybe happens if they adjusti the Stamp Searches and Refreshes. I don't know how to adjust that or if it needs it.
The problem appears not to be a lack of players, but that the Y-Comm system is completely messed up and has problems sorting out what stamps to show to people to notify them that a Max Raid Battle is going on; while it prioritizes people on your friends list, there's no guarantee for it either--people on your list who hide that they're online won't distribute those Y-Comm stamps to people, for instance. I mean, with millions of people playing this game, it theorecially shouldn't be that hard to find 3 people to join your raid.
Super Smash Bros. Brawl had this problem too. It could take up to 45 minutes to fill a room of 4, which is odd considering how popular Smash Bros. has been. By comparison, both Splatoon games could fill a room of 8 in a matter of seconds despite having a smaller player base. Curiously, as SSBB grew older and fewer people were playing online, it became easier and faster to find other players. That likely meant there was a flaw in the way the game looked for players that was more efficient with small numbers of players than large. Though it's different for
Pokémon Sword and Shield, I believe there's something related going on too. The Y-Comm does not appear equipped to handle this many people playing at once, and as a result, the stamps often don't appear until time for Max Raid Battle recruitment has ended, or it just sends the notification that someone's playing it and there's no option to join. (There's also the fact that you see the same stamp regardless of if recruitment is going on, if the room is full, if the battle has already started, or even if the battle has already ended.)
That's why you often can't find anybody for Max Raid Battles. It's because the game distributes notifications to a rather small amount of people, and for some, they won't get that stamp until it's too late to join in.