Do the best audio designers put their ears before the numbers.

Don't know. Haven't heard them all.
I don't even know the design flow for what I have. I know that it has mostly been voiced before offered for sale.

Audio Note is a company that  measures their components to meet certain specs and then they listen. The end result comes from what the designer hears.

There's a good interview with the owner, I'll try to find it.

Not since the days of Peter Walker would any designer say they don't even bother to listen to their design, but I doubt there's even a single one today that relies entirely upon their own hearing. 

Listening is usually only mentioned as a last minute check that some horrible miscalculation hasn't occurred, eg with crossover design etc. Measurement matters, it always did previously, and even more now. 

Advances in measurement techniques and our understanding of how to best apply them will eventually render listening whilst designing altogether superfluous, if it already hasn't done so.
Time is money and the days of designing exclusively by ear have long gone.

Products with serious anomalies will be ruthlessly exposed in today's market. No sane designer will risk that. Even Rega, reluctant as ever to publish data, have had to weather a few storms regarding turntable speed issues. 

Gradually there should be an increasing amount of homogeneity between different designs and products as measurements such as drive unit dispersion increasingly begin to coincide.

Hopefully one day similar loudspeakers will be used to make recordings all over world, thereby ensuring a degree of consistency as the BBC attempted with their designs. Until then some inconsistencies will unfortunately remain.

Audio design has always been a science and not an interpretive art form. The designer's primary task is to build a product which faithfully reproduces the signal fed to it all the way from the microphone to the finished audio file/pressing.

The decision to apply selective EQ/gain is usually up to the listener and not the designer.

In fact the only current exception I can think of is Russell Kaufman who also believes loudspeaker damping causes more harm than good. Apparently, although Kaufman still puts his listening impressions first he also relies upon hours and hours of measurement. 

'At Russell K, we’ve conducted thousands of hours of research into the effect that different crossover types have on the sound from 6db (gentle roll off) to 24db (very steep roll off). We’ve discovered that it’s the combination of drive units mechanical roll off and the crossover working in harmony that produces the best sound.'

http://www.russellk.co.uk/the_concept.php

there is a Nelson Pass interview on youtube, maybe with Steve Guttenberg, where he talks about the combination of measurements and listening...

Listening is usually only mentioned as a last minute check that some horrible miscalculation hasn’t occurred, eg with crossover design etc. Measurement matters, it always did previously, and even more now.

Unless I’m mis-understanding your context..:

No, no no, and no.
Measurement performed in the design and example stages, and in the pre production stages and in the production stages.

But the ear rules the roost. If it sounds bad it is re-assessed via the given electronic tools and so on.

The ear is king for the vast number of high end audio designers. I can’t even imagine the company that is an exception, but I’ll leave room for it.

It cannot be any other way. It would be unbelievably foolish and detrimental to the given audio company to try it any other way.

The ear is ’the decider’.

Ok, re reading your post, I think we are closer to being in a agreement than being apart from it. Listening is not less valuable than measurement. It’s like two identical but differently placed seals and ports on a pressure vessel. One is not more important or less important than the other. One cannot generally rule over the other. More like tasks and aspects that have to, or more should be in agreement - in order to move forward.

But it is possible to have a measurement fail, via the idea of the engineering mind, but still have a sound quality pass or exultation, and have the design move forward to the sales point.

The reason for this.. is we don’t know exactly how the ear works and thus we don’t know how to place what the ear hears across the engineering measurements as a form of comparison and weighting of each in the comparison.

Which is why a hearing positive can cause a product to go forward, even when there is still the idea of a engineering negative on the table. The reality being that we exist within... or shape we operate under or in.. where that comparative and weighting thing between measurement and hearing ----is still not a reality.

And the sensible part, is going for the listening and hearing being the final arbiter, if those considerations about what is heard, is deemed to be strong enough and the engineering 'imperfections' are not severe enough to warrant overriding that hearing based decision. 

Where each scenario is different so it requires human rumination on the entire package, done by individuals, whom are all different.