I'm not so sure the chances are that slim... and I'm not sure that your statement about "some other pathogen's DNA would have to substitute in" means that it's not likely to happen.

How Influenza Viruses Change: Drift and Shift

Influenza viruses can change in two different ways.

One type is called "antigenic drift," which occurs through small changes in the virus that happen continually over time. Antigenic drift produces new virus strains that may not be recognized by antibodies to earlier influenza strains. This process works as follows: a person infected with a particular flu virus strain develops antibody against that virus. As newer virus strains appear, the antibodies against the older strains no longer recognize the "newer" virus, and infection with a new strain can occur. This is one of the main reasons why people can get the flu more than one time. In most years, one or two of the three virus strains in the influenza vaccine are updated to keep up with the changes in the circulating flu viruses. For this reason, people who want to be immunized against influenza need to receive a flu vaccination every year.

The other type of change is called "antigenic shift." Antigenic shift is an abrupt, major change in the influenza A viruses, resulting in a new influenza virus that can infect humans and has a hemagglutinin protein or hemagglutinin and neuraminidase protein combination that has not been seen in humans for many years. Antigenic shift results in a new influenza A subtype. If a new subtype of influenza A virus is introduced into the human population, if most people have little or no protection against the new virus, and if the virus can spread easily from person to person, a pandemic (worldwide spread) may occur (see Influenza Pandemics).

Influenza viruses are changing by antigenic drift all the time, but antigenic shift happens only occasionally. Influenza type A viruses undergo both kinds of changes; influenza type B viruses change only by the more gradual process of antigenic drift.

CDC

The 1957 Asian flu and the 1968 Hong Kong flu pandemics can be attributed to antigenic shift, whereby strains of avian flu and human flu mixed together in swine to form a new virulent strain. The 1957 Asian flu caused a high rate of mortality because the H2N2 subtype of the HA and NA antigens had not been seen by the immune systems of the vast majority of humans infected. When the 1968 Hong Kong virus swept through the world, the virulent H3N2 subtype did not cause as many fatalities, as survivors of the 1957 pandemic had partial immunity to the N2 subtype.

Brown University

There are an awful lot of birds and people out there, so while it may be too early for plastic sheets and duct tape, it's not to early at all for a robust program of research and monitoring. Say, $80 billion for the US and commensurate amounts from other countries.

Can you imagine 2 billion deaths worldwide in the span of a few months? And especially if the victims were, as in the Spanish Flu, young and otherwise healthy? Not quite as bad as Stephen King's "Captain Trips" but devastating on a horrendous scale nonetheless. Like 8,000 simultaneous Thai tsunamis. Or 600,000 World Trade Centers.