Since the US Navy was so worried about its shipyards being blocked by destroyed bridges, how were the Golden Gate, Delaware, and Verrazano approved while the Chesapeake Bay-Bridge was forced to be partially a tunnel?

In part because the USN did not have final say, and while certain of the yards were at risk, they were never the largest or most important at the time of impairment, or the key fleet bases.

For instance by the time the Delaware Memorial Bridge opened the first span in the mid 50's, the Philly yard was already in hard decline as anything but a caretaker for the reserve fleet and occasional overhaul for a larger ship. Her last new build for instance was the party barge Blue Ridge begun in 1967.

And while San Francisco was a very important point of embarkation for the Pacific it was not the only option, and shipbuilding at Mare Island had shifted hard towards subs and other small ships by the 30's and the Golden Gate's construction and it was still designed with Navy needs in mind by being so high.

But other bases, namely the major fleet base at San Diego was purposely kept bridge free for decades until the Coronado Bridge was allowed to go forward, which benefited the Navy in its own way, and Long Beach in LA is literally on the ocean. But in general the danger of being trapped in port by bridges that they had to live with for the Navy was balanced by distributing the fleet. Bremerton could host half the carriers in the Pacific as a home and had clear access to Puget Sound, while other options were taken like a home port in Japan as well. And of course Pearl itself for anything short of a carrier.

On the East Coast this distribution simply wasn't an option. Norfolk by fate and design simply was and is the only real option for much of the force. While the concentration of shipbuilding into fewer and fewer yards made some point moots. By the time the Verrazano was opened and indeed while building, the Navy's presence in NYC was on life support. The Brooklyn Navy Yard had ceased to be a major fleet base decades prior and while still a site for some major builds was hard to justify its expense and location and would be shut down as a shipyard by the mid 60's, while New York Shipbuilding would file bankruptcy by the end of the decade, even building some of the early super carriers was not enough to justify or keep either solvent. While it was also a conscious decision by the USN to close its own in house shipyards and get out of the shipbuilding game, meaning bases had to justify their costs through repair and overhauls or the few legacy contracts and of course faced the axe after anyway, as a sort of proto BRAC this movement really gained steam under the Whiz Kids and McNamara. While Bath and NNS were more than happy to pick up the slack, especially as NNS became the go to guys for nuclear, and while Fore River up in Mass held on as best it could, it too starved eventually.

So we have the Norfolk area through attrition being simply that near unhealthy concentration of fleet base, and private yard with unique capabilities, giving it added gravitas. This can be seen in the Hampton Roads Bridge-Tunnel which was opened in 1959 which allows for guaranteed navigation. And that Norfolk and NNS were in part some of the last games in town was clearly understood when discussions of a fixed crossing began for the Bay.

While we should note that the extraordinary length of the Bay Bridge-Tunnel was a factor as well. When it opened in the mid 60's it was the second largest bridge system in the world, falling only to the Lake Pontchartrain Causeway, so there was only limited precedent for the best way to span 20 miles of water. While a draw bridge like in New Orleans would be unsatisfactory, and raised spans would be needed in multiple spots either forcing an even costlier design, or other unsatisfactory compromise.

So it really comes down to the uniquely powerful political presence of the Navy in Norfolk due to concentration of its presence on the East Coast, and the relative unprecedented scale of the project with few other comparable ones to look to.