Congratulations - the good news is your client likes your work enough to offer a renewal :-).
tl;dr
The bad news is that now your existing contract has expired, your negotiating position is a lot weaker than it would have been if you'd started discussing a rate increase a month ago :-(.
There's nothing stopping you still trying to negotiate a rate increase if you really want to, but you should consider whether it's worth upsetting the apple cart over, or whether it might be better to accept the renewal offer at your current rate and be more proactive at the next renewal.
Long Version
It seems to me like I am at liberty to reject the agreement entirely
You're only legally bound by the terms of any contracts you've already signed. There's good-faith expectations though, and the recruiter might try to put some emotional pressure on you to sign the existing renewal offer, but you're under no obligation to do that if you don't want to.
If you're not sure how you want to proceed just tell them you need some time to assess the situation, but don't take too long to decide because you'll be leaving your client with a lot of uncertainty in the meantime, and that will reflect on you if you do decide to renew.
Junk Emails
The whole thing about junk emails and who said what when is kind of irrelevant, tbh, although there's a lesson to learn about being more proactive about contacting the recruiter in future. Ultimately you should take responsibility for driving the renewal (and knowing when it's due!) in order to ensure you get what you want out of the process.
That might still mean letting the recruiter handle negotiations with the client, but you should be chasing the recruiter to work to your timescales, which might mean, for example, calling them sooner if they're not replying to emails, or even escalating to their line manager if they're not responding.
If you want to get really Machiavellian, you can even let the client know you're having trouble getting hold of the recruiter to discuss the renewal, and if you don't hear from them soon you'll need to start looking for other opportunities. I guarantee you'll get a call from the recruiter the same day :-).
Next Steps
Right now though, you should focus on the moving parts of the situation you're in at the moment:
- You have a renewal offer in-hand at the same rate
- Your current contract has expired
- It might take a while to negotiate a rate increase, and you might be out of work during negotiations
- The client probably won't like being surprised by a request for a rate increase at the 11th hour, and they might refuse it anyway
- They might even find someone else to fill the role in the meantime and rescind your renewal offer
What you need to think about is whether asking for a rate increase at the last minute is worth the risk and the reputational damage it might bring to you personally. If you're looking for a significant increase then maybe it is.
On the other hand, you might just consider that you're happy enough on your existing rate to simply renew for another year with a good client who you're content to be working with, then plan to start negotiations earlier and be more proactive at the next renewal date in a year's time.
Good luck with whatever you decide to do!