Why is potential vorticity so emphasized in technical work, but often neglected in maps on weather websites?
This one has not been answered for a long time and I am going to summarize what I wrote in the comments. If I understood OP's question correctly I believe it is asking why Potential Vorticity  is not shown in weather maps in a operational sense. 

From this old(but still very useful reference) Isentropic Potential Vorticity presuming isentropic surface are available operationally(AFAIK GFS provides isobaric PV surfaces Warning: You need to register to access the GFS link) and you need to interpolate from isobaric to isentropic. 


  Generally, as in constant pressure analyses, one would like to use at least two or three isentropic surfaces to analyze. However, unlike constant pressure analyses, one has to choose different isentropic levels depending upon the time of year. 




| Season     |   Low Level Potential Temperature  |

| Winter     |       290-295 K

| Spring     |      295-300 K

| Summer     |      310-315 K

|Fall         |      300-305 K 



As one can see the correct isentropic surface to choose for a particular day can be quite a challenge. Also each part of the world may have it's own low level potential temperature complicating the problem.


  Isentropic surfaces tend to intersect the ground at steep angles causing analyses near where they intersect the ground to be suspect. One must be careful about over-interpreting analyses made near where the isentropic surface intersects the earth. One would do best to choose isentropic levels which never reach the ground. However, to depict moist/dry tongues associated with synoptic scale systems one tries to go as low as possible without hitting the ground. So there are trade-offs; as usual, there are no quick-fixes!


In my part of the world(tropics) an all season isentropic surface that does not intersect with the earth's surface globally is the 370 K isentropic level. From a isobaric perspective this turns out to be around 200 hPa from a perspective of climatology.

As one can note this is at a very high level and corresponds to the level of the jet stream. As one analyzes lower isentropic levels one can see maps (you need 3-D to plot PV on an isentropic surface) that intersect the earth's surface and when that happens you need sophisticated extrapolation methods to calculate PV especially along orography(and these turn to be generally inaccurate). Over the tropics generally at lower isentropic levels PV surfaces tend to be perpendicular and their utility is restricted to as OP puts it to the inside of tropical cyclones and mesoscale convective systems. 

An example of using PV at lower levels of the atmosphere can be seen in this youtube video Sandy PV filaments