The lighthouse cannot force all ships who use its services to pay. If the ship happens to dock in the port then it can be forced to pay a fee, but ships passing by or docking elsewhere who use the lighthouse do not have to pay a fee. Thus, free rider problem and the lighthouse is less efficient.
Yes, many public goods can be provided by private companies through government regulation. The government must heavily regulate these companies though because they have little incentive to provide cheap and quality goods or services. A private water company operating the only pipeline in town has a monopoly, so you have to buy their water, and they have no incentive to provide water cheaply nor do they have an incentive to provide clean water.
In these cases the private company usually leases the right to provide the good from the government who is the actualy owner/overseer of the good. Why not cut out the middle man? What use is there for a private business if there is no room for competition and improvement from it? Publicly ran facilities usually run better anyway (sometimes not in America do to lack of oversight, no bid contracts, etc. but you have that problem at the top too), with profits going to city hall instead of a business owner who has to turn a large profit on his investment.
You'll also find the government heavily involved in traditionally private goods, such as food. Thankfully government regulation is there so I dont have to worry about samonella in my spinach (how did it get there in the first place?)
