The single-most overlooked, in my experience, is evidence of deaths.
It's a dragon? What's it been eating? Few creatures actually eat EVERYTHING, so what's left by the dragon? Oh, there's an owlbear's beak and claws... there's mangled bits of what used to be +5 plate... (see those runes, there, there and there?) A broken longsword. Dead mind flayers.
And don't forget the massive piles of poo.
Also overlooked: Tracks for Numbers and size! Sure, it's a dragon lair. But its a lot different when the dragon's "wheelbase" is 20x5 vs 80x20...
And don't pull any punches at low-levels. If you're going the route of the "killer GM," go ahead and use that medusa as the boss of dungeon level 1. Served by myconids... (Plants are immune to Medusae gaze). And all kinds of statues... birds fallen from the sky... small critters, big critters... "Hey, isn't that Fred the town Bully? So that's what happened to him... hey, there's my missing purse!"
My suggestion:
Mirror PC stats, abilities and numbers to give them a challenge. If you have a fighter, wizard and rogue group, and you want to put them up against monsters, you'll need to set them against something tanky, something ranged, and something that can hit hard and fast.
Alternately, you could design special monsters like a Colossal Scorpion. Give the torso the tanky stats, and make each pincer a rogue type damage dealer. In addition, the stinger could be considered limited ranged (30'/60'). This would give the scorpion 3 distinct parts that can be targetted, as well as a main body that will kill the whole thing.
You can apply this to any set of monster enemies as well. An Ogre, axe throwing orc, and orc shaman would be a good trio to set up against PC's if you balance their stats according to the PC stats.
The problem you're going to be facing with trying to balance this is something you've already pointed out, the CR ratings don't mesh with single fights. It will take a good amount of experimentation to get a feel for it, but if you start with PC stats and incorporate those into monsters, you should be fine using that as a baseline.
Numbers version of level 6 characters:
Fighter - 65 HP - 2 handed weapon
Wizard - 32 HP - evocation
Rogue - 40 HP - Whip, throwing knives
Creature battle:
Colossal Scorpion +7 to hit (Mutli-attack, this creature attacks with two claw attacks and a stinger attack)
Legendary Resistance 1 - Once per encounter, this creature can turn a failed save into a success
Regenerate - As a bonus action, the scorpion can sacrifice 40 Torso HP to fully regenerate a destroyed claw or stinger. That claw or stinger only becomes available for use on the creatures next turn.
Torso - 150 HP - damage resistant non-magical
Left Claw - 40 HP - Pincer attack 1d10 + 3
Right Claw - 40 HP - Crushing Grip 1d10 +3, auto-grapples
Stinger - 40 HP - Acidic spray, ranged 30'/60' 1d8 + 4 acid damage
If you kill the claws and stinger, the scorpion dies. If you kill the torso, the scorpion dies.
That's just one suggested battle to keep things interesting. The main point is that a battle like that enables a monster to take as many actions as your players can, and it doesn't destroy the creatures ability to remain a challenge.
If you break the enemy down into trios as well, consider giving them similar abilities to player characters. This will ensure the challenge rating is appropriate. Try to avoid making the mistake of jacking up their AC, to hit bonus, HP and resistances. Keep it consistent with what the PC's have to maintain the arena feel to the tournament.
Best Answer
Give them a better idea of the threat level
If you are playing a sandbox style game, it is up to the players to determine what level of known risks they wish to face. Thus, you probably do not want to telegraph too strongly that you think they are ready; that decision is up to them. Dice have vagaries and players play with differing levels of tactical skill and system mastery and character optimization. Unless you plan on using a deus ex machina to put your thumb on the scale for one side or the other, you only know how deadly an encounter is meant to be, not how deadly it will actually be. So, in a sandbox style game, let them figure out how much risk to take. Just give them a fair idea of what those risks (and possible rewards) are.
In your dragon example, they can meet someone that can tell them more about that specific dragon's capabilities and thus judge for themselves if they wish to face it. One indirect way to do that is to tell them more about the last adventuring party so they can say "Yes, we are more powerful than that now".
Put the encounter on a timer
One way to encourage the encounter is to do something to make it urgent. If something bad happens if the issue is not resolved within a time limit, they will be much more motivated to find a way to resolve it now. From any but the most ruthless or most dedicated to a "breathing world" GM, this will also be a not-so-subtle but also not meta-gaming signal that the GM thinks the encounter can be resolved within that time period.
Perhaps the dragon is now threatening a town by demanding a major payment of tribute within a week or it will attack. The adventurers now have a deadline to resolve it or the town will be directly harmed. They won't even be able to say that they can get the tribute back from the dragon later when they are yet more powerful, if much of the tribute is in the form of consumables like cattle that will be eaten. Or, if you want to be more cliché, perhaps the demanded tribute includes a hostage.