But surrly anyone searching for an Advance would find the walk up fare at the top of the list as a cheaper fare?
How utterly bizarre. Why would one ever be sold? It wouldn't show as the cheapest single or return.
Whilst flexible tickets may be cheaper, they are not always the most prominent fare on screen. Not all websites display results in the same way and the myriad of fares, along with singles versus returns, can makes things very difficult for people to follow, especially those who don't travel often.
Some people only look for Advance tickets because they have been conditioned by the industry into believing that is how they'll get the best price, so may not notice that they are paying more for less.
At work, I was once supplied with an Advance ticket outbound and a full price single coming back, at greater expense than a flexible return. The person who booked it was absolutely adamant they had booked the cheapest option, even though they hadn't.
Linking back to the non-TPE example given by Starmill, currently, an attempt to make a booking for this Thursday from Crewe to Wolverhampton using the 11:22 outbound in standard class is £35.90 for an Advance *Single* - nearly twice the price of an Off Peak *Return* and almost as much as an Anytime *Return*. Coming back on the 22:15, if choosing First Class, the Advance *Single* is £68.00 - which is actually more expensive than a First Open *Return*. Anyone daft enough to select that combination would end up paying £103.90 for a highly restricted mixed class return, at far greater expense than any flexible ticket. If they miss their train, they may also be forced to buy one of those cheaper tickets!
I've been shouted down for this in the past, but I do not believe Advance tickets should ever be available for the same price or higher than the relevant valid flexible ticket. That should also include the sum of the parts. i.e. An Advance each way may be cheaper than a flexible single each way, but shouldn't be sold if the combined cost is higher than the relevant flexible return. I understand some websites will warn you in such circumstances, but others don't.