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One could argue that Satoshi has intentionally "lost" access to the coins that they 've mined.

Even in that case, the public key is valid. Which means, that there is a matching private key, which can be used, to sign those coins, and move them. But fortunately, the Genesis Block is unspendable, so even if you know the key, you cannot move those coins. But still: you can move all coins, which were deposited later into this P2PK, or into matching P2PKH.

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There are assumptions, but no proof whatsoever, that Satoshi mined many blocks (~20,000).

There are only traces of extraNonce, pointing out, that some miner produced a lot of blocks. It doesn't have to be Satoshi, and the estimation of 20k blocks is also covered with a huge assumption, that a single miner own all chunks of nonces (which may be incorrect, because every time, when extraNonce starts again from zero, there could be another miner, with similar mining power).

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The coins that they 've generated haven't moved either.

Some of them moved, but then, they are excluded from the pattern. For example: http://satoshiblocks.info/

We have blocks from 0 to 14 with one extraNonce line, and people assume, that Satoshi mined them. But: block number 9 and 12 are explicitly excluded from the pattern, because those coins were spent.

And then, we have blocks from 15 to 25, forming another extraNonce line. And people assume, that if the first line was mined by Satoshi, then the second line also was.