Whenever the demand is greater than the capacity, some transactions will inevitably be delayed.
Whenever the demand is greater than the capacity (supply), then the transaction price increases and those who will pay more get their TX executed, those who dont will have to wait a bit more.
Thus you can still send your TX in normal amount of time, you just have to pay more to get it cleared.
Yes, those transactions that pay higher fees will go first.
However, the capacity of the network is ~0.8 MB (~1500 trasactions) every 10 minutes. If there is 1.6 MB of transactions (~3000 tx) in the queue, half of them will miss the next block,
even if each of them is paying 1 BTC of fee.
Fees can only decide which transactions will get served first; no matter how high they are, they cannot reduce the
average waiting time -- not even by 1 second.
Moreover, it will not be "a little longer". If clients (all over the world, independently of each otehr) are issuing 3000 transactions every 10 minutes, at most 1500 of them will be processed in the next block. The other 1500 will go into the queue, adding to the backlog; and will be confirmed only after the traffic again drops below 1500 tx/block. Even then, if the demand drops to 1300 tx/block, the the backlog will be cleared only at the rate of 1500 - 1300 = 200 tx/block.
So, a surge in demand that lasts a couple of hours can take many hours to clear. A large percentage of the transactions will be delayed until the backlog is clearing. Even if most transactions are paying high fees. The average time to first confirmation, which was ~10 minutes before the jam started, will then jump to several hours.
Even if tools like replace-by-fee (RBF) and child-pays-for-parent (CPFP) are available, they will not reduce the average waiting time by a single second. If you use RBF to push your transaction to the front of the queue, so that is gets mined in the next block, some other transaction that was expecting to be in that block will be pushed to the second-next one, and another that was to be in the latter will be pushed to the third next block, and so on.
If your transaction got into the backlog (and there is a significant chance that it will, no matter what fee your smart wallet chose), there will be thousands of other clients, just as impatient and wealthy as you, that will try to get their transactions ahead of yours, using those same smart wallets and fee-adjusting tools too. So, after you sent your payment for your shuttle flight, or to reserve the last room in that hotel in Paris, you will have to keep checking the queues, increasing the fee of your transaction, maybe for hours -- only to wait until the backlog is clearing, anyway. With a high probability, you will pay a lot more, only to wait much, much longer.
If the average demand is greater than the capacity, the backlog of unconfirmed transactions will keep growing forever and will never clear. A fraction of the transactions will fall off the end of the queue, after sitting there for hours or days, and will never be confirmed. The average waiting time will then be infinite.
To summarize:
If the average demand is much less than the capacity, there is no fee market: all transactions that pay minimum fee will be processed in then next 10 minutes.If the average demand is close to capacity, it will be greater than capacity during peak hours or days; then a growing backlog will form until the demand subsides, and the average waiting time will shoot up from 10 minutes to hours.If the average demand is greater than the capacity, the backlog of unconfirmed transactions will keep growing forever and will never clear. The average waiting time will be infinite and some transactions will never confirm.Any bitcoin developer who wants to see bitcoin develop a "fee market" is either incompetent, or does not care about its users.