If your game is small but you can catch the interest of users enough, you can make them install a propper Java.
(So a well made propaganda site with nice screenshots will help)
But this wont work for like a small Packman clone.
Packaging the JVM with the game really only makes sense in large Games, not for a 2 MB casual game.
Make it easy for people to get to the official Java Download link.
If they already bothered to to download an installer/exe (so not just a browsergame)
they might not bother to download/install Oracles JVM.
I think Minecraft also helped push the ecosystem of normal gamers installing and accepting Java.
So thanks for that.
I think getting users to run a fitting JVM ist less the issue than providing them a convenient way to start the game itself.
(exe, maybe even with an installer/desktop icon)