If towing is controlled by the political apparatus, then perhaps they don't send the tow trucks if a vehicle is blocking the bridge to make a political statement.
You need to spell out exactly how you think that would work. Because the only "profit motive" I'm seeing is that the bridge operator would want to avoid losing money on refunding tolls, and would be incentivized to provide towing services. But isn't that bridge operator exactly the same government transportation authority that you are proposing doesn't need to be involved in providing towing services? Or are you proposing that the day to day maintenance and operations of a piece of critical infrastructure should be privatized, and then subject to provisions that ensure it's a risky/bad investment?
The point is that the bridge makes sure it has towing capabilities on standby for the fastest turnaround possible, since the operators of the bridge only care about the bridge. If a city government has a standing contract with some towing company, they probably use it for all sorts of things, so you wouldn't have that immediate standby capacity.
I don't like the work "privatized." Instead, this would be a bridge built by entrepreneurs and operated as a business.
Too bad. The meaning fits.
It's unrealistic to propose that a new bridge would be built by private money. There's plenty of private investment money floating around in the Bay Area, and it hasn't come close to happening yet. And you have completely failed to respond to the concern that your proposed toll-refunding mandate would make a privately-operated bridge an even less viable investment than the non-starter it evidently already is. You're just dogmatically asserting that the free market will always provide, when it looks pretty clear that the free market would just say "no thanks".
So, I'm not sure a "profit motive" is really the solution here. Yes, the bridge operator could make that contractual offer and add towing infrastructure out of the goodness of their heart, but there's no incentive to doing so. A decent local government does have that incentive (as they are responsible to the commuters as voters), so they offer that service. You could have regulation that forces the bridge operator to provide this service instead of the government, but is that really less government intervention?
"Alternatively, the bridge operator could be legally mandated to offer a refund of the tolls if the transit of the bridge exceeds a contractually-guaranteed time."
I know it offends the libertarian sensibilities, but there's no reason for the operator to voluntarily do that.
On the other hand...
The Fort Lee lane closure thing. (https://en.wikipedia.org/wiki/Fort_Lee_lane_closure_scandal, https://www.nytimes.com/2016/11/04/nyregion/george-washingto...)