Yes, they do begin to realize how much of an asset the public servants are, but it can take half a term, or even a whole term for them to build the kind of trust required to work with them effectively.
Are we properly organized for our public servants to work well with political appointees and elected officials? In some ways, I think the government is organized against that. The departments are so complex, so large, and so rich with resources, it's hard for elected officials, who are often thin on staff, to move the government any particular direction. This gets even worse when an issue requires capabilities across territorial government departments, which is increasingly the case, at least in the area I know: national security.
I think most public servants recognize how silly it all is and how they are in many ways set up for failure or at least a lot of difficulty in implementing whatever initiative it is. However, elected officials and the public often default to blaming the public servants as a whole, rather than realizing they are the ones with the power to reshape these institutions, and public servants would like that as well.
However, we need to be realistic. While we have a vision for a new generation of public service, change has to begin with small wins, under present conditions, that attract people toward a new way of doing things. Change by fiat will not really be change. Elected officials can give the effort a push, but they get so sucked into urgent day-to-day things -- "firefighting," politics, the news cycle, crises, new events -- that this is often dropped as a priority. I think it's great that so many people are beginning to raise awareness about this.