That's primarily because, in contrast to staying in San Diego, a move to Carson will take years, and result in a ton of lawsuits against the Chargers by angry fans, angry cities, and angry former business partners. In 1998, the City of San Diego passed Prop. C, which got Petco Park built. The campaign to build Petco was all-emcompassing, and probably one of the better political campaigns run. Per the terms of Prop. C, Petco was supposed to be built by 2002. It wasn't. Thanks to a lot of litigation, the Padres didn't open Petco until 2004. But it did open, thanks to favorable rulings by local judges who may or may not have been swayed by local sentiments. By the end, most of the plaintiffs in the cases against the Padres were denigrated.
But if the Chargers move, the local judiciary, along with the local jury pool, won't be so inclined to give the Chargers a break. What's more, the proposed Carson stadium is being built in direct competition with a sports complex that's already in Carson. So, not only will you have a lot of sour grapes from San Diegans, but there will be at least one lawsuit from people in LA with fairly deep pockets. Maybe the NFL convinces them otherwise, but maybe these deep pocket interests side with Stan Kronke's Inglewood stadium. Either way, the Chargers will be tied up in lawsuits for years.
But lawsuits aren't the only problem - all projects have to undergo an environmental review under California law. These EIRs generally take 18 months if there's no problem. The only way out of such a study is if the California Legislature specifically exempts the project from an EIR. Now, if the City and the Chargers figure out a stadium deal, then, most likely, the Legislature would pass such an exemption. But if the Chargers move to Carson, then such a deal is unlikely because the current Speaker of the State Assembly represents the City of San Diego. Oh, and unlike the Qualcomm site (where the Chargers currently play), the Carson site is rumored to be highly polluted (the Qualcomm site has undergone a major clean-up), so the EIR would not go smoothly.
All this means is that the Chargers' proposed move to LA would take years. Where in those years would the Chargers play? It can't be in San Diego, their popularity would be almost nonexistent. Could the Chargers afford to play in a terrible nonfootball stadium in LA for five years? What if in year three of the move the stadium plans fall through, the way that so many other stadium plans have fallen through in LA over the years? Where do they go then?
So, I guess my point is that the Carson isn't the escape hatch the Chargers make it out to be. Leaving San Diego and ending up in limbo in LA is the likely scenario for such a move. And with that being the case, the City and County shouldn't feel pressured because of the threat.