I totally agree about level scaling, HOWAYVER, as a D&D DM I've seen firsthand what happens when the players end up in a low-level area you expected them to tackle earlier.
Make it a race to get the macguffin before the level-scaled anti-party does.
I think I'd put it like this:
Dungeons / Adventures A-Z are available. Many are level 1, many level 2, etc.
Antiparty chooses where they will go, and party chooses. If there's no spying, each won't know what the other party is doing. Maybe they accidentally hit the same dungeon. Figure out some table of how many rooms the antiparty gets through, chance of success for party level vs. dungeon level, and you note which items were taken and which monsters slain by the anti party. If the player party comes by later they will walk through some empty rooms that the antiparty cleared, similar to how the antiparty enters a dungeon partially cleared by the player party.
Instead of random encounters with another random adventuring party, you track hour-by-hour turns if there are two parties in the same dungeon to see if they cross paths. It's possible the player party will go down one hallway, adventure several hours, and return to find that someone else has come through and cleared the other hallway while they were gone.
The top-level work is making up a bunch of adventuring parties, the success tables for party level vs. dungeon level, and rolling to see where each party goes. Once that's done, you might have no NPC parties in the dungeon with the PCs - in which case, you can leave the rolling for their success for lull periods such as players dividing treasure. You really don't need to know how the other parties did until the players return to town and hear about their exploits - success or failure.
This also opens up interesting inter-party interactions:
It's not just that "several adventuring parties have gone missing near the Scorched Caves", but it's specific parties like the Reavers in Red and Coldwell's Delvers. Maybe something to do with their party composition was a problem - not enough Clerics to fight the Undead, or their wizard specialized in cold magic and the creatures there are also magically cold?
An NPC party has been mostly destroyed. Only the Thief and some hirelings survived. Will some party add the Thief to their group to get his intelligence and experience with that dungeon? What if the Thief's story of how his party died isn't entirely true?
Another party finds a cool magic item that the player wizard really wants. Will you attempt to purchase, or steal it, or hire someone to do your dirty work?
Your party finds a relic that's needed by another party. They don't want to tell you why. Maybe it's the last piece of a thing that unlocks a treasure - or an ancient evil! Will they try to steal it from you if you won't sell? Will they try to steal it just so you don't suspect its importance?
An NPC party's Fighter is high enough level to build a stronghold. He's calling adventurers to help clear out the wilderness around it while his group supervises construction.
A new adventuring party has arrived from the Utter South, made up of cloaked monks who refuse to speak, led by a mysterious wizard. Why does he look suspiciously like the necromancer you chased out of the graveyard?
The dungeon you cleared - and vacated - has been claimed by some other adventuring party, who came up with a legal deed of antique origin for the property. Why do they want the place? Did you miss something in your previous excursions? Are you willing to trespass and investigate?
The dragon comes by every decade to accept the offering that the townsfolk give it - one virgin. Now that the town is full of adventurers, are you all willing to just let some innocent soul be spirited away to an awful fate?
The swamps harbor mischievous sprites and druids who waylay travelers and steal from caravans. But any individual attempt to enter the swamp gets distracted and misdirected, ending up back at the perimeter or else drowning in muck. One adventuring party wants to gather a massive force to sweep the swamp and, with superior coverage, defeat the swamp-dwellers.