I will speak an insider of customer services, not only working in a customer services center in having to deal with so many others. Because there's such a bulk of customers, in order to provide efficiency they will work to what's called an 'SLA' or a service level agreement. This can often mean you won't always get the same person and because customer services isn't exactly a high paid job, you're not getting the best of the best, you're not necessarily going to end up with somebody on the ball and training quite frequently isn't that brilliant either.
Therefore the service you get can be inconsistent and you may be unlucky and get the guy who is good enough to keep his job or is able to look good when it comes targets and scores, but not necessarily on the ball service wise. But at the same time, you can end up with the well experienced guy with a decent brain between his/her ears.
Customer services as a result can be frustrating. But because I know how call centers work, I know when having to call other call centers at work, I know how to get what I need. If I get a bad agent, I try again. I won't use any of the people I deal with at work as an example, because it would be inappropriate, but I can use a recent example of customer service I experienced. I use a host called Heart Internet, I was having a problem, the guy I got was extremely knowledgeable on my issues, because I was going through some fairly technical stuff and he nailed it - I was impressed. But the last reply I got was from an agent who clearly didn't understand the issue and completely got it wrong, but I already had enough to figure it out, so I just thank him and closed my ticket off. If I needed further assistance, I may have requested assistance from the original agent, reopened my ticket or open up something like a web chat or make a phone call to ensure I keep the same person.
It is possible the guy I had finished work for the day or was on break or had something else to deal with. And I can think of my own supervisors saying to us at work, "that email is about to go out of SLA, could you take a look at that". I may have gotten the guy who knows a lot about technologies like My SQL and PHP but little about ASP.NET. People typically want a quick service, but it can sometimes affect quality - customer service center have to find the middle somewhere.
I can see your guy is labelled as "Support Tech", my suspicion is these are two different guys. Both having different understanding of company policy and processes. Don't get *why* they would ask you to write it down, but then I have dealt with idiots in my line of work, so it doesn't surprise me too much (fortunately, it's not as bad in our office - we are very good at supporting each other). I can understand the need to ask for proof, as we sometimes ask for it in dealing with customer cases - not to accuse people of being dishonest, but to cover our own backs should be people be dishonest. But we'd consider something hand written to be suspicious more than anything (why not just take a picture of the code on the case?). But this wouldn't be the first time I've heard of people misunderstanding policy and procedures. I can list a few names of call centers I've dealt with off the bat for it.
Best course of action? Would it be to go the legal route? Citizens advice bureau? Watchdog (or whatever you have)? Trading Standards? Bad customer service isn't illegal and it is possible there's something in the terms and conditions you've agreed to and possibly something you've not read that covers their back. Unless it is determined that Valve are breaking the law, chances are they won't get involved.
Instead, try again, get a different agent, if you find you don't have success, try requesting an escalation to a senior - like a line manager or supervisor, usually they tend to be more consistent. Getting irate or even making threats (like in the ball park of Trading Standards) is more likely to make it more difficult for you. Suppose it depends, but I think the typical mentality of a person working in customer services is not, "we have wronged this person and we will do all we can to please them" or "they're our customers, therefore we have a duty to fulfill", though that is the typical expectation, but they're most likely people who don't want to be there, who don't like their job and aren't necessarily feeling particularly good will do the guy trying to make their day harder. Your typical customer services agent hates their job. Also, just because you've paid for something doesn't suddenly mean you're entitled to be given an A-grade service and what they owe you is the terms you've agreed in purchase - but few people know what they agree to because they are selectively blind to terms & conditions and like to click "yes" to the box that says "do you agree?". Even if you pay a lot of money, you might have invested that money for a product, but not a service.
I don't get the mentality of "I paid them, they owe me" because yes, you've given them some of your hard earned cash, but they've also given you something in exchange. All they owe is what was agreed. If folks have a problem, they need to read their agreements, granted, it's time consuming, hence I accept the consequences if I don't get my own way over something I didn't read. If I'm paying a lot of money, *then* I will look me deeply into the terms of my guaranty...heck, I read my guaranty before buying my PC. But I don't read any of my Steam ones, to be fair, I'm prepared to take the risk.
Maybe try their complaints procedure, I know for where I work complaints are handled by a completely different group of people.
I know it's frustrating and know it's bureaucracy, but it's the society we've made for ourselves. We want quick and efficient service with minimum cost to ourselves. You want a company to invest more in their customer services, you will not see the money paying for it coming out of profits, but it will come from the price tag. It's why when it comes to the 'Apple is too expensive for the technology' debate, I will throw in the line that whilst it's not perfect, they do invest more in their customer services.