I’m going to reiterate the above answer - relax, you absolutely do not need to do any volunteering or extracurricular activities beyond reading. I’m currently studying PPE at Oxford, and I never did any. The PPE admissions process is entirely academic; the admissions tutors will not care at all either way about how much or how little volunteering or work experience you’ve done, not least because they’re very aware that access to these opportunities is far from equal for all applicants. As far as reading suggestions go, just follow your interests; it doesn’t particularly matter what you read so long as it’s relevant to PPE in some way. The first year theory of politics course is largely centred around the ‘classics’ you mention, but you don’t need to worry about actually starting to read them until the summer before you start the degree if you end up getting an offer.
Remember, the personal statement has a very tight character limit, so, especially as you’ll probably want to mention all three subjects in it, you really won’t be able to discuss loads of books. There’s no need to fret about reading a ton; in fact, what you’ve mentioned almost sounds like it’s enough already. At the end of the day, though, the personal statement is one of the least important aspects of your application; whether you get an offer will be far more determined by your interviews, your TSA score, and your GCSE and predicted A-level grades.