What you're describing are tanks that gear for STR without meeting the HP threshold. That's simply poor gearing on the tank's side. They will be putting a lot of faith in the healer's hands and have a minimal margin for error that can easily lead to wipes. That's no good.
What Technole is vouching for is gearing STR after the HP threshold is reached. You say you're comfortable with an 11k HP tank, what if the tank has 15k or more? If they don't dip below 4k HP, they function identically to a tank with 11k. Only a precious few skills work based on the percent of total HP, like Thrill of Battle, Stoneskin, or Benediction. Most other abilities are based on the amount of healing or damage coming in. This means that, as stated before, an 11k HP tank will use these abilities to the same effectiveness as a 15k HP tank. If those abilities are sufficient for an 11k HP tank to survive, any excess HP is just fluff for the tank. It provides no meaningful value if that HP is never used.
Likewise, the failure to use self healing or mitigative abilities will be equally difficult to handle regardless of whether one tank has higher HP. If the incoming damage is higher than your HPS, there's little you can do about it regardless of the tank's HP, except hope the DPS can clean things up before the tank goes under. If you're dealing with a big pull, chances are that's unlikely. Higher HP may provide a few seconds of cushion, but it's not going to save a bad tank from making bad decisions. That's just poor tanking. Typically, if you're having a bad time healing a STR tank, you'd have a bad time healing them VIT spec'd. There are good tanks that gear STR and good tanks that gear VIT, there are even great tanks that swap accessories between pulls and and stance dance too. These are tanks that have developed an intuition during battle much like how we healers have. They know the flow of battle, when they will be taking heavy damage, and when it will be less severe. Pay attention to these tanks, how they use their abilities to not only make life easier for their healers, but simultaneously have the ability to output decent DPS as well. One does not have to come without the other.