how those domains prices get establish?
There is nothing to understand. Registries decide the price of their premium domain names based on what they want and how much they think they should cost that is how much they think people are prepared to pay for them (technically, for a registry, any single name is exactly like any other, it means that the cost to run the system, per domain name, is the exact same amount for any domain name basically - one could argue that the most known ones will tax the registries nameservers more but that should remain a minimal impact due to DNS caching -, so premiums are just a way for the registry to manage their list of domains and try to get more money out of some of them). It is the exact same problem on the "second" market, how do you judge the price of any given domain name? This is far from being objective.
I am sure that whatever domain you will buy today at some price, in one year, you will believe that if you sell it you should earn more money than what you paid today, because it is "known", it has traffic, etc. This case is not so different with registries having premium domain names.
isn't this price establishment goes against free-competition and illegal-setting-prices
You are using big words here. No one forces you to register a specific name, you can choose any other. So why would it be against free competition? You are free to compete with any single name you want, that has no consequences on prices for all other names.
And why would it be illegal? Based on what law?
Each registry is the sole entity dealing with domains in its TLD or zone, and decide if it wants to give them for free, for a standard price for all or for different prices per domain based on local and subjective criteria.
gTLDs are covered by ICANN agreements. You can see a lot recently about .ORG or .COM and how the contracts are changing to allow more liberty for registries, at least for the one before the 2012 opening. All the new ones after that date are already basically free to use whatever pricing they want, they are only required to announce (to registrars) any price changes (do note that it also happens that some domains change category going from standard to premium or the opposite).
For ccTLDs it is more complicated, as each government decides on rules, and you will find less premium in those cases. But it can happen too.
Also do not forget about registrars: they will put their own margins, which may be higher for premium names. But that can explain why you see "similar" prices, registrars can only work with the registries prices (there are sometimes discounts and things like that, but certainly not for premiums).
The subject of price comes regularly, maybe my other answers can also help:
TL;DR: even if that is not the answer you would like to hear, in your case my best advice is: "if the name you want is not available - including not available in the price range you want to pay for it - just stop trying to go around this fact, and just choose another name, in the same or another TLD, this is the fastest and simplest path".
I may understand the price of a .bank for 750$, maybe, or domains with only one registrar, even hotels for 65k$ seems stupidly expensive.
Some TLDs are restricted to some eligible entities only. High prices can be part of the deterrent package. However, you seem to mix both standard
price (which registries are free to choose, it can be from $1 to hundred of dollars, excluding premiums of course), and specific premium pricing.
(prices from https://tld-list.com/tld-categories )
The categories you see here are purely arbitrary. They were not started through this scheme. And of course you do not see ccTLDs.
The prices then shown are from some registrars, I do not think you can conclude anything on how many registrars were polled (registrar prices are far from publicly available in a standard format for all cases), data freshness, and other things. Said differently: this is just one data point among others.
We (a just born startup) came with a great name
You can always try to contact directly a registry and asks if they can do anything for your case. Sometimes it happens. For example right now
.new is not available for the general public, but might be available for some ".new advocates" that the registry decides and who will have a domain. Again, if you are sure of yourself and have time to spare (and will accept a lot of refusals or no replies at all) you can contact the registries directly.