No, we very much did try brinkmanship. That's was Theresa May's "No Deal is better than a Bad Deal" rhetoric was all about. You lapped it up even as we were criticising it. But it was, and will remain, a mistake to pursue this line. The EU won't be "brought to heel" because we just shout a bit louder next time.
I agree that No Deal wouldn't last but I am not sure that you would get any more of a "sensible" deal, in your definition of it at least, after trying No Deal for a while. If, as is almost universally expected, the fallout is more damaging for the UK, then who do you think will have more incentive to end that situation? What ends up happening from a No Deal is in fact far more likely to be even worse for the UK than what we have already been offered (and, of course, worse still than remaining).
If No Deal is indeed our bargaining chip, then it is the only one we have. We shouldn't ever use it; as soon as we do, we will be massively worse off.