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Sure, the bridge operator could do that, but why? Bridges don't operate as efficient markets with heavy competition, instead they tend to act as regional monopolies. If you don't cross this bridge, you usually have to drive an hour or more out of your way. This market position of the bridge operator gives them outsize leverage over their customers, and so there is no incentive for them to compete on things like price or low traffic congestion, because who are they competing with?

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?