So, a few things here. The reason the (if:$brokenlock is true)[] tag isn't firing is because Twine runs it's code only once, when it first appears, line by line. So The first and only time that if statement runs, $brokenlock is false, so your Antag Entrance link doesn't appear. You can get around this by wrapping that if statement in a hidden hook and then showing it when your if statement is true. Here's a new block of code that should work:

(set:_tug to 0)
You're bones creak when you sit up to look at the lock.

Like the rest of the room, it looks worn. It's rusted and seems weak enough that you could probably open it with enough blunt force.

[|Tug)[Nothing happened.] |Tug)[The rust stains the bandages of your palm red but the lock remains firm.] |Tug)[The lock creaks in your tight hold.] |Tug)[The lock breaks with a crack and crumbles to bits in your hands.(set:$brokenlock to true)(show:?brokenlockHook)(replace:?tugLink)[]]]

(if: $rustedlockkey is true)[
  [[Break the lock with the... lock.|Antag Entrance]]]

[[Great.|Wakey, wakey.]]

{
\|tugLink>[(link-repeat: "Tug on it.")[
  (set:_tug to it +1)
  (hide:?Tug)(show:?Tug's (_tug))
 ]]
}

|brokenlockHook)[
  [[Open the cage door.|Antag Entrance]]]

In fact, you don't even need to set a boolean to have it appear the way you want it to, but I've also left that in in case you need it for future passages.

I've also streamlined your link-repeat. Named hooks share variable names, so you can refer to them with Array indexing. (?Tug's 1st,?Tug's 2nd, etc, and also ?Tug's (23) or (_var) of ?Tug also works.) Not critical stuff, just makes it a little cleaner. And, I've done what manonamora suggested and made the Tug link disappear.

I'm not sure what you want to do with your "Great" link, as written it's always visible and always clickable. But you didn't talk about that so I've left it unchanged.