"If bracing doesn't contribute "much" to the sonic signature of a guitar, what does? "
IMO, most of the character of the tone of a guitar derives from the size: big guitars are more 'bass balanced'. Much of what I think of as the quality of the tone, how good or bad a guitar it is given the size, depends on how well it's made. This is not simply, or even mostly, a matter of precision workmanship, but rather whether the various elements are well balanced and work together. Thus, for example, getting the top thickness and bracing 'right' tends to produce a more even and clear tone. Note that what is 'right' for one top may not be for another, which is one reason it's difficult to make consistently good Classical guitars in a factory. Things like materials and finish are far down the list of influences IMO.
"Various guitars with similar specs; spruce, rosewood, scale length, body shape and dimensions, etc. What makes them sound different?"
It's instructive, and humbling, to try to make a 'matched pair' of guitars that actually sound the same. I've tried several times, using wood that was cut 'in flitch' for the parts, with measured properties that are well within measurement error, carefully controlling the thickness, mass, and resonant pitches of all the assemblies, and finishing them together so that they are as much 'alike' as two wooden objects can be. They still don't sound the same. In the most recent case, a friend looked at spectrum charts showing the response between 50-1000Hz, and commented that if they were electronic devices, you'd say they were the same. But in 'blind' listening tests audiences could readily distinguish them. They were, of course, very similar in sound. Audiences expressed no particular preference in average between them when asked.
The problem is that there are always small differences in the materials and construction that are difficult to measure and impossible to control fully. Even two slices of wood cut next to each other from a very large tree will vary somewhat in the exact distribution of stiffness and mass from one place to another. These small differences don't alter the way the guitar responds in the low range, below 600-800 Hz,, say, but do have increasing effects as you go higher. The way the different parts of the guitar work together to produce sound diverges more and more, producing measurable, and audible, differences. Your hearing is very sensitive to these sorts of small changes in the high frequency range. You are, after all, the descendant of the people who could hear the tiger sneaking up through the bushes, and he's trying to be quiet.
"What gives a certain line of guitars a recognizable "house sound" that transcends materials?"
In some sense you can think of building a guitar as being a complex problem with a number of major variables. The 'correct' value for many of these is not a fixed thing, but rather co-varies depending on what the other variables are: they are 'codependent' variables. It sometimes seems to me that if there are, say, a couple of dozen of these, you can make a very fine instrument if you get something like 20 of them 'right', whatever 'right ' is in the particular context. What's interesting is that it almost doesn't matter which ones you concentrate on; if you get those 'right' you'll end up with a guitar that most people will say is 'good', and the others can be pretty much ignored. The 'house' sound is then a function of which things the maker chooses to concentrate on getting right. You can use a thicker top, as Hauser did, or a thin top, like Ramirez, and end up with an instrument that Segovia would find satisfying. They will sound different, of course, and may prove to be better adapted to different sorts of music, but both are still good instruments.
All of this is, of course, informed opinion. Take it for what it's worth.