1) It misses pricing. If I am next to another gas station or hot dog stand, I have competitive pressure; my best strategy is to price just below the competitor. If I am isolated, I can jack up prices by however much people are willing to spend to not go further.
2) It also assumes just two players on a beach. With more players and a longer beach, you end up with everyone spread out. With three players, the two outer players will continue to move in towards the center player. If the center player ever has less than 1/4 of the beach, they'll want to move to just outside one of the other two players, who will then want to move to the center.
Customers will naturally migrate to where there is more options, similar to the network effect. See: mall food courts, food trucks, strip malls, etc. So it's possible that the two clustered stands end up splitting a >50% share of the market.
This is actually a common misunderstanding. The example of ice-cream vendors on a beach or, in this case, gas stations, are toy examples.
The model is not really about location. It is about product differentiation in SOME dimension. By extension, it is also about the number of market participants with differentiated products.
Distance or location stands for this differentiation of products.
This is typical of econ models. They are not trying to be realistic, but show a certain mechanism.
By the way, the model can be extended with different pricing mechanisms and many actors (for example on a circle, not a line, to show that generally too many firms enter a market than can be supported).
As for your points:
1) Pricing depends on demand and maket configuration. Usually, one would work with what is called the "inverse demand", where the company has basically solved price as a function of the the amount of goods it WANTS to sell, given other market participants.
Note how pricing is implicit in this. While these functions are of course usually assumed to be well behaved, you could do what you want here. So it doesn't miss pricing, it just abstracts from it
In particular, that you can jack up prices depending on location is _explicitly_ in this model.
2) Note how the locations depend on the distribution of demand along the "line". Your intuition depends on this as well. If demand is concentrated somewhere in the middle, you will get a different result...
By the way, think about what would happen if you could differentiate in two dimensions? You'd be on a plane or a circle on which a demand distribution exists... etc.
The assumption in the example is that the product being sold is a commodity, which cannot be differentiated significantly in terms of pricing or quality.
It limits its applicability, but it doesn't mean the model is incorrect.
"In Hotelling’s Location Model, firms do not exercise variations in product characteristics; firms compete and price their products in only one dimension, geographic location. Therefore, traditional usage of this model should be used for consumers who perceive products to be perfect substitutes or as a foundation for modern location models."
1) It misses pricing. If I am next to another gas station or hot dog stand, I have competitive pressure; my best strategy is to price just below the competitor. If I am isolated, I can jack up prices by however much people are willing to spend to not go further.
2) It also assumes just two players on a beach. With more players and a longer beach, you end up with everyone spread out. With three players, the two outer players will continue to move in towards the center player. If the center player ever has less than 1/4 of the beach, they'll want to move to just outside one of the other two players, who will then want to move to the center.