After all, the last Satoshi wouldn't be mined until around the year 2140, long after we're all gone.
Sure, but this question will be become pertinent long before then. Average fees earned per block is somewhere in the region of 0.1 - 0.2 BTC at the moment. By the 9th halving, the block subsidy will be 0.09765625 BTC, which will mean it will be less than the average fees. This will happen around 2044, so only 20 years away. Another few halvings after that and the block subsidy will be ~10% of the total fees. If fees aren't enough to maintain the security of the network alone (as will be the case in 2140), then it is very unlikely that fees +10% will be enough to maintain the security of the network, which is scenario which will be reached in the lifetime of many (or even most) current bitcoin users.
I think the combination of the increasing value of bitcoin coupled with most smaller transactions moving to layer two, resulting in the base layer being used predominantly for larger transactions and channel opening/closing and therefore larger fees being acceptable, will be enough for fees alone to maintain the security of the network.
There is some very interesting discussion about this in the following forum thread and on the dev mailing list:
"Surprisingly, Tail Emission Is Not Inflationary" -- A post by Peter Toddhttps://lists.linuxfoundation.org/pipermail/bitcoin-dev/2022-July/020665.html