Are you sure? For a single journey to be £4, you're assuming that "the vast majority" of passengers changes at least once. I'm not sure that's true - Fairly certain it's not for "within" an urban area here. And two journeys with changes (i.e. 4 journeys) already brings you over the day ticket price.
I think it will vary based on the reason for the journey. If your destination is central to the town or city you are going to and / or you are prepared to walk up to 15 minutes then it's rare to need to change. However not many work places are in the centre of cities these days, not many cities are pleasant to walk through at 7am, and not everyone can lug heavy bags that far.
Part of the reason I don't use the bus much is the need to change and the corresponding cost and inconvenience. OT, but much of the inconvenience is poor connections leading to much extended journey times, but cost is still a factor even if the connections are seamless. This is why I think the London system needs rolling out country-wide where there is either no or just a small extra charge where you connect to a second bus within a certain time period.
With reference to an possible national Oyster style system, work is already in progress on a nationwide tap and cap ticketing system for buses. Project Coral, instigated by Midlands Connect, has support of all the main bus groups, most Local Transport Authorities and is funded by the DfT. There is also involvement from the various payment system providers.
When I first went to Taiwan, over a decade ago now, I was very taken with their Easycard system. Valid on the majority of public transport nationwide(1), aside from long distance rail and HSR (2) and can be topped up, and more importantly the credit used, at the numerous 7-11, FamilyMart etc type corner shops. Different card types were available for children, students, pensioners etc so the system knew what fares to charge. Made getting around really cheap and easy (3), and made me wonder why we didn't have something similar in the UK.
The fact I'm still wondering 12+ years later must say something about this country.
The card is now available as an app as well, although I believe you can only register if you are a Taiwanese national. Japan has similar, although there are two or three competing ones rather than a single universal card.
(1) - With no apologies to the CCP
(2) - I believe it now can be used to buy tickets on both rail and HSR, but you still need a ticket for HSR.
(3) - Aside from the buses where they have a complicated system of whether you tap when getting on or getting off (or both) which changes part-way through longer routes, and where the sign telling you which to do is only in Mandarin.