Why bitcoin is not full p2p? The goal of bitcoin is to serve as p2p electronic cash. But it still needs a 3rd party (miners) to approve/authorize/record/track/store transactions. The difference is this authority is decentralized and distributed.

Why is it not a complete p2p (like gold, fiat cash) electronic cash? Is it not possible to create such system with cryptography? If yes what are the difficulties?
it still needs a 3rd party (miners) to approve/authorize/record/track/store transactions

Role of miners vs other participants
In the beginning miners were not a third-party, every wallet owner was also a miner, all peers performed the same roles.
Nowadays mining has become a separate activity but miners do not really have any special control over the bitcoin network. They have a specialized role but I believe their influence over consensus rules carries the same weight as ordinary wallet nodes. Its just that a mining business deploys a lot more nodes than the average wallet owner does. In any peer-to-peer system, even a completely homogenous one, one person or organisation could deploy a large number of nodes.
approve/authorize/record/track/store transactions
approval
All nodes approve transactions, if all ordinary wallets disapprove of blocks from a miner then that miner loses money.
authorisation
There is no authorisation in the Bitcoin network protocols or consensus rules so far as I know.
The only action similar to authorisation is the P2P process of proving ownership of private keys in order to prove the right to spend the outputs of a transaction. Every wallet does this.
Recording transactions
Miners are responsible for recording transactions in the blockchain. So they do have a special role here. Bitcoin relies on no single organisation having more than 50% of the total available hashing power.
Tracking transactions
All full-nodes track transactions (i.e. keep local records of transactions in blockchain blocks as well as maintain an in-memory pool of unconfirmed transactions). Miners have no special role here.
Storing transactions
All full-nodes store a complete list of transactions from which blocks of fully-spent transactions are optionally pruned. Miners have no special role here.


Why is it not a complete p2p (like gold, fiat cash) electronic cash?

Gold is not P2P in the modern world. Generally people who buy and sell gold directly do so in the form of coins or ingots - the production of coins and ingots is performed by a limited number of businesses not by the buyers and sellers. Often it is governments that mint gold coins. Often governments regulate the gold market. Gold assayers are often relied on by buyers.
Fiat cash is not P2P. Banknotes and coins are typically created by governments, not by buyers and sellers. The amount of fiat cash in circulation is typically managed by governments. Inflation rates are set at the whim of politicians.