The thing you are missing is that the attachment doesn't require that the weasel grapple the opponent, nor does it affect the weasel's speed or ability to move in any way. Rather than ending the 1 hp/round drain by running away from the weasel, you likely must chase down the weasel and initiate a grapple. Once you've managed to catch the creature, its -12 grapple modifier makes removing the attatchment relatively simple. Depending on the ontology of animals as a detail of setting, this may look something like the following:
Case study: Druidic Animals
Where people can talk to animals with e.g. Speak With Animals, animals are moral agents dedicated to nature and balance and stuff, have goals and culture and relationships, and are on average significantly wiser than humans, elves, Dwarves, etc.
3 level one commoners v.s. 1 tactically-savvy weasel in a dense forest.
The weasel hides in a space with a tree connected to the canopy and in a patch of heavy undergrowth until a commoner enters the space. The Weasel takes 10 on this Hide check, resulting in a Hide total of 26, which the 1st level commoners cannot reach. When the space is entered, the weasel attacks in the surprise round, 1d20+4 v.s. a flatfooted AC of 10 for a 70% success rate. If it hits, it attaches.
Hiding while attacking imposes a -20 penalty, so the commoners now have a good chance of spotting the weasel (about 30% each), and they can probably locate the bugger via listen, since it's now making noise. Nonetheless, the weasel has about a 9% chance of still being undetected, in which case it waits the turn out, hides again on its initiative, and waits for the bitten commoner to die (8 rounds, or 48 seconds, later). Even if the other commoners decide to premptively attack the space in which their friend first started taking bite damage from a hidden source once per turn, the weasel benefits from cover (total AC 14 for a 70% miss chance for the commoners, 51/50 if they both attack the space) as well as separate 50% and 30% miss chances, totalling in a 17.85% combined hit chance for a thorough search (if they hit the weasel, it's probably dead). If either of them are stupid enough to actually enter the weasel's square they are subject to an attack of opportunity that is equally capable of applying attachment just as above (but with less eyes to see the attacking weasel).
Assuming it is spotted as it attacks, the weasel then 5-ft-steps diagonally up onto the opposite side of the tree and out of the undergrowth, using its climb speed, and uses the tree's cover to hide again (at the lower total of d20+11). Everyone rolls initiative. The weasel wins approximately 30% of the time, and goes before the guy it bit around 60% of the time. The commoners can't run or charge, but they can still possibly move to within 5 feet of the weasel within a turn, maybe make their spot checks, possibly by some of them using Aid Another, and then maybe hit the weasel with their 1d20 v.s. 12 AC attack (40% odds of a hit once the weasel is spotted). If they expect the weasel's position, they can also just attack the space and hope to hit it (accepting the 50% miss chance and probable cover bonus), resulting in a 51% survival rate for the weasel if they choose the correct side of the tree to attack from (so as to negate cover) and all attack before it flees.
The weasel, if it is alive on its turn (either because the commoners all lost initiative or because those who didn't failed to find and hit it) withdraws using the withdraw action into the forest canopy, where its climb speed can allow it a higher effective movement rate than the commoners, and moves to set up an additional similar ambush further along the commoner's path of travel or eat the fallen commoner or whatever as fits its goals.
In a setting with more anti-animal bias, the weasel might stay adjacent to the commoner and continue attacking, or attempt to flee less effectively, or have set up less good of an ambush, or just run away and never attack humans, or something.
A dire weasel functions similarly except that it is less good at hiding, can't climb, has enough hp to maybe take a single hit, and doesn't need to enter an opponent's space to attack. It's also, obviously, scaled up to challenge higher-level characters but that doesn't really affect its combat role.
Hopefully the test cases help show that weasels are extraordinarily dangerous for their CR. The only ways to remove the attachment they cause, which otherwise results in swift death for low-level characters, require that you catch them first, which is not a simple task either because they are tiny and can climb (for regular weasels) or because they run at a 40 ft. movement speed (for the dire variety). However, weasels also have almost no defensive prowess. Dire and regular weasels are essentially very-low-level high-level D&D characters: they turn combats into rocket tag at levels 1 and 2. If the weasel attaches and gets away, the thing it attached to is dead. Even at high levels, attachment, while difficult to pull off, can kill almost any opponent when used by a regular weasel. Undead are immune to a whole lot of stuff, but 'bite damage' isn't one of them so e.g. a Lich can be temporarily killed by a single weasel benefitting from the 1st level Druid spell Magic Fang if it fails to kill the weasel before the animal escapes (perhaps due to a lucky application of the 1st level Druid spell Entangle).
This, of course, is somewhat different than how weasels work in real life (though surprisingly accurate to weasel folklore), and so you may well wish to change this all to something more... reasonable.
Speaking of weasel folklore, I'm going to leave you with a passage I ran into while reading about real-life weasel burrowing habits in the above linked source:
The seemingly innocuous little creature we call the weasel is an insatiable killer driven to murderous frenzy by a large parasite residing in its stomach. It sucks the blood of its victims, conceives through its mouth and gives birth through an ear, can squeeze itself through a wedding ring, and magically changes from brown to white within hours of the first snowfall each winter. (See the different varieties of weasels in the image gallery).
which I think is pretty great.
Unfortunately, Blood Drain does not work the way I've been running it for years, so my actual play experience with the Dire Weasel is quite limited (i.e. nonexistant). My experience with the regular weasel, however, is quite extensive.
Lastly, a note on PCs fighting weasels, though your question is of a different bent: weasels have very low hp and low movement speeds, making mundane ranged attacks one of the most effective ways of taking them down. While NPCs like commoners and warriors are unlikely to have or be able to afford the luxury of a short bow, 1st level parties may well have a character with a light crossbow or even a full-blown archery-focused PC. Enemies like the weasel, with it's use of cover, concealment, and difficult terrain, coupled with its extremely low hp and ability to present a credible offensive threat to the party provide a rare opportunity for mundane ranged characters to shine. Of course Magic Missile solves the encounter trivially, if the weasel can be found (even Magic Missile has a miss chance against an opponent with total concealment), and Burning Hands kills all the weasels in a fairly large area without allowing much of a chance of failure, but magic is sort of always better and at least in this case it's much less better than normal.
As a warning, though, weasels are extremely dangerous and there is a significant chance that a PC or even several will die if they fight a weasel, so don't introduce them to your game if PC death from goblin dice is something you don't want to risk happening.
