The other place to look would be to the W-SW of the airport, in the already-mostly-industrial area bordered by the Caltrain tracks, the San Thomas Expressway, and either SR 101 or the single train track that runs near De La Cruz Blvd.. The only problem is, that's no San Jose, that's Santa Clara (see ), and since "all politics is local", that doesn't count.
The area SW of the airport that is in San Jose is the empty-looking strip between Coleman Ave and the Caltrain tracks. A chunk of that has construction in progress, another chunk is Avaya Stadium and associated parking, and I think the rest is reserved for a future BART terminus and associated stabling sidings (similar to what BART has at Millbrae).
On the other hand, though… maybe that could still be used for development? It would be expensive, since any developer would have to maintain an empty shell on ground level, but it may be doable.
There are policy and technology options to work around the tightness of the city and the airport. This is a city that needs its airport. It's a rare asset to have such a versatile and productive airport so embedded.
That's why it will ultimately be replaced -- it's too close and is limiting growth of of that same downtown core.
I will also point out that if you develop in a properly dense, urban fashion, you can create a large amount of space with relatively low-rise buildings.
I am skeptical of the premise that the airport is limiting downtown development when I can see huge swaths of land devoted to asphalt parking lots and single-story buildings at present. You could redevelop/build new square footage for decades just on that.
San Jose airport's runways are aimed right at San Jose's core downtown area (near Diridon station) where all of this new development is taking place.
Google wants to build near Diridon where Caltrain already goes, and BART will eventually be extended there (or at least to nearby San Jose downtown station)
Being transit friendly requireds high density, high-density means tall buildings. San Jose can't continue its low-rise development if it wants effective transit.
All of the routes have too many turns. Maybe some are unavoidable, but splitting up into a east west route on the North end, a North south route on first Street / downtown, and whatever makes sense in the south of downtown routes would add transfers for many trips, but also reduce slow turns.
In the downtown section, it stops at too many intersections.
In the downtown section, one direction goes through a paved pedestrian area, which it needs to travel at walking speed while laying on the horn to encourage pedestrians to get out of the way.
+1 about SJC being outstanding. It's so damn easy to get in and out of. I really like flying through there. SAN is also pretty good, if you're on SWA anyway.
There'd probably be a decent amount of economic collateral damage by moving all that air traffic out of SJC and into SFO/OAK.
Also given SFO's sky high airfare prices, further consolidation will not help residents wrt that cost line item in their lives.
One downside though would be it would add to the already crazy traffic in the 101-237 corridor.
It felt like planes were barely passing over the roof until I made the approach in the air and felt like it must have been several hundred feet of separation.
Still closer to a building than any other approach I can recall, but never felt dangerous.
It certainly does reduce the margin for error though.
SJC, while super convenient (I live in the South Bay) isn't a premier airport and will never be. Half of it is old and needs remodeling.
We need housing and density so much more than we need more flights out of SJC.
Obviously we also have to require aircraft to take off to the north, yes?
Let's see how we can implement this proposal.
SJC has two parallel runways: 30L/12R and 30R/12L. The runways are 700 feet apart.  
We could operate the airport like a two-way highway where you only drive on the right. We permanently close runways 30L and 12L, and allow takeoffs only on 30R and landings only on 12R. So you approach from the north and take off to the north.
This will result in oncoming traffic 700 feet to your left, whether you are taking off or landing.
Passengers may get used to seeing other jetliners zoom by them at 250-300 miles per hour. After all, we do something like that on the freeway every day.
As a pilot, my worry is the wingtip vortices, affectionately called "wake turbulence". These are little cyclones that come off the end of each wing, travel with the wind, and can flip you upside down! The two-way traffic will assure plenty of opportunities to get caught in one of these.
OTOH, the oncoming traffic and wingtip vortices may only be a problem on a very calm day.
On many typical days of the year with moderate winds from the north-northwest, we may have to close the airport for landings and only allow takeoffs. Conversely, when the winds reverse due to weather, we will only allow landings and cancel all takeoffs.
Perhaps this plan is not quite as simple as it first seemed.
 Runways are numbered by their approximate compass heading divided by ten. A runway can be used in either direction depending on wind conditions, so it has two different designations that differ by 18, or 180 degrees. When an airport has parallel runways, they get an L or R suffix for left or right. So runway 30L and 12R are the same physical strip, at an approximate compass direction of 300 degrees if you are approaching from the southeast, or 120 degrees when approaching from the northwest. Similarly, the parallel strip is runway 30R and 12L.
 SJC runway map: https://upload.wikimedia.org/wikipedia/commons/a/a6/SJC_Airp...
So you approach from the north and take off to the north.
I also mentioned that crushcrashcrush's plan would require closing the airport to "opposite direction" traffic on all but the calmest of days: you would have to allow only takeoffs in normal winds, and only landings when the wind reverses.
My comment was a tongue in cheek attempt to point out the problems with the plan by taking it seriously.
I did forget to mention one other issue: even if you required approaches from the north and takeoffs to the north, it still wouldn't allow you to build taller buildings south of the airport. Airplanes need to be able to go around (abort a landing and take off again if the approach isn't right), so in that case you would be taking off to the south.
Also, you approach based on the wind.
you can only approach SJC airport from the north, not the south
On one occasion, I flew via SFO. I vowed to never ever do that again. From that point forward, I always flew via SJC, and I was much, much happier. The airport was clean, reasonably up-to-date, and never had major traffic problems that I saw.
I would be very unhappy if that airport got shut down or crippled just because some people want to build tall buildings in the area.