There's definitely economic value but it really only goes to the companies building facilities, the suppliers, and the hospitality industry immediately surrounding the events. In an already well-developed city, they aren't going to be building many new permanent structures. Many will be temporary that will eventually be torn down or just converted back to what they were originally. The money is spent, distributed to the companies that directly participated in the build up and run of the Olympics but there's little gain made after.

A more interesting way of doing the Olympics would be to only allow for developing countries to participate in the selection process. Each country would be required to meet a certain level of funding to guarantee they can support the entire Olympics. Foreign investment would be encouraged. There would be a requirement for some aspects to be permanent construction, you couldn't just build a tent city for the athlete village. Then a name is picked randomly. The host country then receives major foreign investment, not just in sporting arenas, but in many areas of its economy. The Olympic committee could also collect dues from participating countries based on GDP that would go to the host country for economic development. It would basically create a lottery system for the economy of developing countries. The build up to the Olympics would create the infrastructure needed for future investment. This would likely require host cities to be selected much further out in advance. An oversight committee would observe the development and if milestones are not met, a host city from a developed country that does have the infrastructure necessary would instead be chosen.