It doesn't matter what "Satoshi" intended. On-chain transactions are scarce because the Bitcoin blockchain is designed to be deterministic (this is self evident) and to be highly distributed (this is also self evident). Miners mint coins and process transactions, non-miner "full nodes" verify the integrity of the network (they are the only 'clients' that don't have to trust that the nodes they are talking to are behaving properly, using the desired protocol).

Increasing block size increases network bandwidth requirements (higher cost & quality required), compute/memory requirements and storage cost & complexity (UTXOs). Larger blocks increase the cost of running nodes. It is not known exactly how increasing block size affects the composition of the network, but the general argument that increasing block size reduces the number of possible participants in the network is obvious. Given that on-chain transactions carry a high cost (relative to suitable off-chain solutions), it is obvious that not all transactions warrant on-chain settlement. Therefore, an off-chain (L2) architecture and implementation are needed in order to 'complete' the Bitcoin architecture so that it can reasonably support real-time, lowest-cost payments. Bitcoin is still in 'beta', in with regard to low-value payments. Lightning is the spearhead for implementing a payments layer for bitcoin. You can check the status of Lightning on mainnet here: http://lnstat.ideoflux.com:3000/dashboard/db/lightning-network?refresh=5m&orgId=1

Enlarging blocks (pricing more participants out of the network) purely in the interest of lowering fees in the short term so we can lure more people onto a Beta network doesn't seem productive. It is better to start by (1) optimizing blocks so bits are used more effectively (SegWit), then (2) maturing a layer 2 implementation, then (3) reviewing block size increase requirements once the actual demand for on-chain transactions becomes clear (after all the low-value transactions have been moved off-chain and usage patterns with the L2 settle into patterns we can use for planning purposes).

But can they now? I'm not sure we have a problem that needs solving.

No. They just don't know that they care YET. They will care. A lot.

The providers of wallet software will learn to adjust the UI so it displays a denomination by default that is appropriate to their users. The Satoshi is a sufficiently small unit for the foreseeable future (even if BTC were $1,000,000.00 each a Satoshi would be $.01). The market reporting and exchanges will adjust similarly. We can help them along by talking about "mBTC" amongst ourselves and in public, rather than BTC. We can make suggestions to the exchanges, publications and UI developers as well, but they will find their way even if we don't push.