It's absolutely both

I think the only thing I'm really stressing is, sensitivity only matters as much as it indicates how much work the amp needs to do to get similar tonal and volume output into the room. An amp need to work 4x as hard to produce volume when you drop an octave. Whether you want to think of it as needing to add lows to a more midrange forward speaker, or cutting lows to the TC, getting a similar tonal profile on the TC means you have freed up the amp's wattage for general volume without running the amp harder, while the TC wasn't far behind in the first place for general sensitivity. As Ken pointed out earlier, if you want to utilize the TC for the unique voicing it affords us (which would be intensely difficult to achieve with a lot of other speakers), you'd probably want a capable amp. But at that point you're not talking about getting comparable performance volume as with a more conventional speaker, because that speaker wouldn't be able to do what the TC does at volume with less wattage cost to the amp.
I'm just making sure I spell out my perspective on this for anyone who has sensitivity concerns, so maybe some of it might seem a bit redundant to you as I'm sure you have serious technical knowledge and experience in application with this stuff. I know that there's been similar sensitivity concerns with the fEARful, and I just want to be careful others don't misunderstand what the TC does when it's a medium sensitivity speaker that simply doesn't get the apparent volume bumps in either upper bass/low mids or upper mids where wattage wouldn't cost too much to amend.