I don't know if and why he said that, but obviously it is wrong in the long term.  When the bloc reward becomes insignificant, ALL proof of work has to be paid by fees.  That cannot be "insignificant", because if it were, it would be easy to attack the chain with more PoW: it would only cost you more than the "total amount of insignificant contributions".

The principle itself of security by PoW comes from the fact that the total amount of value wasted (in the end, by the users) must be so significant, that nobody has any incentive to spend MORE to attack bitcoin.
This collective bill of bitcoin is paid by all the bitcoin holders, right now mostly due to inflationary pressure (inflation tax), but in the end, by those transacting.  In a certain way, this is *fairer*, because the cost of bitcoin's PoW is a service to those doing *transactions*, while with coin creation, all bitcoin OWNERS pay the bill (some kind of socialist redistribution: owners pay for the service to transactors) ; with fees, the transactors pay for themselves and the non-transacting owners are not affected (no inflationary pressure).

However, that bill is important, and for that bill to be important, transactions have to remain scarce.

So there is essentially NO WAY to make transactions cheap and fluid on a finally PoW protected chain without inflationary remuneration (as has been the case until now and in the decade or so that follows).

EVEN if you take transactions off-chain, they are STILL only secured by Proof of Work, and they will have to pay the bill IN ANY CASE or bitcoin becomes insecure.  No matter what.

This is simply because remunerated Proof of Work for consensus decisions was a BAD IDEA in a sound money coin.  (and sound money itself is an economically bad idea too, but that's another discussion: it turns the asset into a speculative asset falling into a deflationary spiral and makes it non-liquid as a currency).

Satoshi's claimed quote would make sense if the "cost of mining" were related to "the amount of data".  But it isn't.  The resource cost of data and networking is INSIGNIFICANT as compared to the ON PURPOSE wasting of resources in proof of work, which is not related to computing resource usage, but to CHAIN SECURITY.

As such  "the size of the transaction" doesn't matter AT ALL in the security model of bitcoin's proof of work.  It does, if the market squeezer is "block size".  But that's a silly squeezer.  As I said, the problem cannot be solved.