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I know how transaction fees are calculated. Also I understand that a transaction with a higher fee will be approved much faster than a transaction with the lowest one.

Does the bitcoin protocol allow committing a transaction without paying a fee (even if the transaction will be approved within 2 or 3 weeks)?

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The protocol allows it but I believe many nodes will not relay a transaction whose fee is below a configured threshold.

See

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The default minRelayTxFeeRate is 1,000 sat/kvB. Nodes using the default value will not relay transactions that pay a smaller feerate than that. I just did a search for confirmed transactions paying less than the minRelayTxFeeRate (feerates up to 0.00000249 BTC/kWU, excluding coinbase txs): enter image description here

Lately, most days there are a couple and never more than a handful. Given that there are over 62 blocks worth of transactions tracked on mempool explorers offering feerates below the minRelayTxFeeRate, it's safe to assume that those few transactions were created by the corresponding miner themself and included in their own block, and there is no miner that includes below minRelayTxFeeRate transactions from the mempool without out-of-band encouragement (or there would be a lot more). enter image description here

I would not expect any transaction paying less than minRelayTxFeeRate to get organically confirmed even when blocks are not full otherwise.

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Yes you have, fee is optional. When the mempool clears some miner will likely mine your transaction

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    These days most nodes on the network, and miners, will not accept or forward a transaction with a fee below the minrelayfee (which for recent Bitcoin Core versions is 1 sat/vbyte by default). Perhaps if you send it to a miner directly, they may include it, but even that is unlikely without adding some incentive (in the form of a fee...).
    – Pieter Wuille
    Commented Aug 22, 2021 at 12:55

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