I think a lot of this is tied up with other people moving and also the schools question.
The first thing that strikes me is your house is fine. You have 4 beds and 2 bathrooms and 3 loos and a garden. You have a very decent family house and are not in the position lots of people who move out are, which is being in either flats or very small houses. You like your area and it works for DH getting to work and you like London. These are all good reasons to stay.
People might move for reasons that don’t really apply to you...their houses are too small, or they have shed loads of money perhaps which mean they can buy a vast pile in the country, or because their kids won’t be able to do well in the schools they are likely to get.
Schools is the issue to really think about. It sounds like you have good schools available to you locally and you are happy with them. Don’t look for reasons to be dissatisfied. If you are happy with local schools you are very lucky. Lots of children will do really well at such schools. Check the data for high/middle/low achievement in entry and see if there are plenty of children similar to the profile of your kids and see what the results are like. See if the school uses some setting. Yes, Comps will always be more mixed. That is true, but lots of kids do really well in them. Consider if being in a more mixed environment is a problem in your eyes or is a positive. Some people are horrified at the idea and want to cosset their child, but lots don’t. Many people who are truly comfortable in London embrace diversity, but those who aren’t happy with diversity often move out. If you are happy with the schools, your house and everything else sounds like you’re good to stay.
I’ve known people move out for schools. Some had children who were academically weak or had special needs that they wanted met. They felt their kids might sink in a large Comp and that they wouldn’t get into the selective London schools, so they moved out for a wider range of less selective schools for their particular children who they thought just wouldn’t get into a school they liked and would work for them. If your children are robust and doing well, and if you’re not interested in independent school or state grammars, you can avoid all the 11+ conversation and be happy you are in catchment for a good school. If you need to do more research into local schools to reassure yourself they really are okay, then do it. Speak to some current parents too and not just those with kids your age who have already decided the schools won’t suit them.
And as you know yourself, it’s easy to feel that when lots of others seem to be doing something, you can feel you’re missing out or missing something. It is a case of working out what matters for you and identifying where the reasons they want to go don’t apply to you. You don’t sound desperate for a huge garden....unless you want to do lots of gardening or think the children will spend all their time out there, you’re right that open spaces like parks are better anyway for playing, especially beyond when children are small. If you don’t yearn for country life, but enjoy city life and walking everywhere rather than going in the car, then staying sounds right for you.
You are actually very lucky. You already have a decent family house and good local schools, a walk to work and enjoy city life. You don’t NEED to move. You don’t really sound like you want to either, but if you do want to, it sounds like you can. And you’re right to consider it now because with a 9 year old, time flies and if you’re going to do it it’s better to do it in time to apply for senior school. Unless it’s always been part if your plan, or you love Surrey or dislike London, or the schools are a major issue, you sound like you’re in the right place, but only you can weigh all those reasons people go (and stay) and work out which apply to you.