From what I understand, the smallest fee you can pay to miners per transaction is 1 Sat/byte.

It's not true. A transaction can contain a fee of any amount. There is nothing to stop you from creating a 250-byte transaction with a fee of 1 satoshi, or 17 satoshis, or even 0 satoshis (no fee at all). Of course, it will depend on market forces as to whether miners would actually decide to include such transactions in a block.

Many wallets ask you to choose a transaction fee in satoshis per byte, but that's just a convenience for the user (since most miners look at the ratio of fee to transaction size in deciding what to confirm). It's not an inherent feature of the blockchain. Under the hood the wallet just multiplies this number by the transaction size to determine the absolute transaction fee. You could patch the software to set an absolute transaction fee directly, or perhaps even just enter a fractional number of satoshis per byte.

(There are currently rules preventing low- or no-fee transactions from being relayed over the peer-to-peer network, but that's not the only way to get your transaction to a miner; also, those rules are not part of the blockchain consensus rules, so they can be changed without a fork.)

It's true that if we ever get to a point where 1 satoshi is worth a lot of money, then a transaction would have to have either a large fee (>= 1 satoshi) or no fee (0 satoshi). The former would be expensive, and miners have no incentive to confirm the latter (though they can if they feel like it).

There's been lots of discussion about what will happen if 1 satoshi ceases to be small enough. One possibility is a lot more use of off-chain transactions. Another is a hard fork to add more decimal places, which would presumably be uncontroversial. See Will we ever need smaller amounts of Bitcoin than a Satoshi?