I live rurally and grew up rurally, but spent 20 years in London as an adult.
There are manay, many more things on tap in a city for teens, but they tend to be centered on inner cities, not suburbs anyway, as you say.
Rurally, there is NO public transport (where I live, there is a bus every 2 hours, and they stop at 5pm!), and you must be prepared to be a taxi service to get to swimming, cubs etc, as they're all in the next town/ village along. That bus service sounds dreamy! If it connects to a reasoanbly large town, that'd be great. Our nearest town worth a night out is 2 hours away by bus.
There is virtually nothing for the later teens to do.
That said, they make stuff of their own. A youth bus that visits weekly is well patronised, we have a library that does a lot ( it did Manga drawing classes over the Summer) and as there is one primary, one secondary ( takes the hassle out of school selection!), the friends are all local and bikeable when they're older. But, if they don't like the school, the other is 20 odd miles away.
A lot of the teens are outdoorsy, as that's whats about. There's a big off road biking scene and lots of running clubs.
9 is a good age to move.
What worries me is that you "long" for a rural life. It's not a bed of roses, villages are often poorly resourced, it can be very lonely initially, and very cliquey. You can't get away from people. And the countryside isn't always accessible: we live in a massively farmed area, and there are less footpaths than when I was in London! And all the newbuilds in our village are TITCHY, with next to no gardens. Go for ex-council 1950/ 1960 builds for gardens ( we have one, with a 250 foot garden out back, but new builds in the country are no bigger than elsewhere). Look VERY closely at the development plans for your area, the East of England in particular is set for massive expansion, and this won't show up on housing searches, you have to look at the areas overall regional development plans.
Also look at employment in the area: rural areas where I live are some of the worst in the UK, but around Cambridge it's fine, but VERY expensive. You need to really, really, research.