I would like to respectfully suggest that you change your approach.
the authorities must give me every chance to get one within a reasonable time limit or produce other satisfactory evidence of your identity and citizenship...
Later, you say:
...it is not my fault and I should deserve a compensation as I am gonna miss the flight and have to pay extra 90 pounds for the renewal (new process).
The first misstates your experience, and the second is incorrect.
First, airlines are required to return to the departure point at the airline's expense (although the airline may try to collect the fare from the passenger) every passenger who is refused entry at a destination or transit point. Thus, airlines are extremely careful to check every passenger's documentation; airlines use the Timatic database for this purpose. Because there are an almost infinite number of combinations of factors (which countries, what kind of entry or transit, passenger citizenship, passenger residency, visas held, age, etc.), reference to a standard database makes sense.
If the airline thinks you won't be admitted at your transit point or destination, you will not be allowed to board the flight. You can see your own document requirements by entering your data into a Timatic portal; here's one, there are others as well. Put your data into Timatic: you'll see that with your current documents you can fly from the UK to Greece, but you'll be barred from flying back to the UK.
The airline will rely its Terms & Conditions of Carriage, to which a ticket holder (whether a purchaser or only a user) is deemed to agree by presenting themselves as a passenger. Without exception, these T&Cs provide that it is the passenger's sole responsibility to have the correct travel documents to make the flight. This is not the airline's job, it is yours.
Second, the government (whether of Greece or the UK) has no duty to keep track of what documents you hold, what kind of condition those documents are in, or when they expire. This is, again, your job alone.
Here, you didn't pay attention and left passport renewal and Greece national ID replacement until very late in the game. While the time required was increased because of a error by the Greek bureaucracy, in the real world...things happen. It would be delightful if the Greek bureaucracy responded by accelerating your Passport renewal, but bureaucracies are not known for timely and moral responses. A more prudent traveler would have presented the application earlier, in case an unexpected issue arose.
You can complain and whine about how poorly you've been treated, but it won't resolve your issue of insufficient documentation to re-enter the UK, and it will certainly not help you grow into a more capable traveler.
An effective reaction, on the other hand, would be to follow the good advice given by @Willeke and @gsamaras and @Chris Melville in their Answers. Reframing your experience as "I screwed up this time, but I'll pay more attention in the future so this won't happen again" would be a much more productive strategy.