Congratulations - the good news is your client likes your work enough to offer a renewal :-).
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 :-(. The client and the recruiter both know any rate increase will take a while to get approved (if it even gets approved), and you'll possibly be out of work and losing income while that happens.
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're not sure how you want to proceed just tell him you need some time to assess the situation. (But don't take too long to decide because you'll be leaving your client in 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 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 them 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 the rate increase request 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 consider is whether asking for a rate increase now is worth the risk and the reputational damage it might bring to you. 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, and plan to start negotiations earlier and more proactively at the next renewal date.
Good luck with whatever you decide to do!