From the standard player's prospective: Your game needs a lot of work and with all the money you make off of skins and mystery gifts it shouldn't be difficult to hire enough people with the right knowledge to fix some of the issues LoL has. We buy RP not because Super Teemo is on sale, but because we love this game, we love you as game creators, and we hope that every dollar we spend with make more of this magic happen for us and our friends to enjoy.
I obviously don't know your experience in industry, but, this comment in your post indicates a lack of knowledge of how hiring processes work and how project in general work. There is the concept of the Mythical Man Month (good book by Fred Brooks - check it out) that throwing more people at a problem does not solve the problem and, often, causes more issues.
Also, I (and I'm guessing almost nobody else) buys skins in hopes that my money is spent on a specific issue. I trust Riot to spend the money properly when I do spend it, but I spend it because...well...skins are awesome.
I want to play this game. I want to love this company and I do not want to say anything negative about you wonderful people who work at Riot. You make it difficult though because all you do is elude and deflect. The more you can tell us about what keeps you from fixing bugs and making new content quickly the more patient we will be.
I partially agree with you here. Fixing bugs in software (I work in the software industry) is not an easy thing. Confirming that they actually are bugs is also challenging. I suspect that Riot as a company doesn't want to just come out and say, "Yup! That's another bug." It may not be! And, once they have determined the problem is a bug, they want to determine a solution (or at least a correct path) before revealing more details. And, on top of all that, it takes a lot of time and effort if they were to keep everybody updated with the status of everything in these forums. That is time they could be spending doing actual work. But, I do agree with you that Riot is often very vague in their communication. This is frustrating to me. I would rather hear nothing at all.
Why are the servers having so much difficulty keeping up over the last few weeks when you have a very good idea of how many players to expect?
Why does LoL have fewer servers than other, less established MOBAs?
The servers are being actively attacked. And when I say servers, it may not actually be Riot's servers directly. It may be upstream of their servers (which has happened) and they don't have direct control of that. Go read up on DDOS at Wikipedia. DDOS are notoriously difficult to protect against. In addition, how do you know how many servers Riot and all the other MOBAs have? Even more importantly - more servers does not mean better service. Network topology, links, available bandwidth, and a plethora of other things come into play.
Why, for example, has Smite (released at the end of March) done more re-works and VU's than LoL has in four years?
Because...they're new. When you're small, it's very easy to be agile and move quickly.
Why does it take so long to fix bugs that the community finds for you?
Most bugs have a lot of interaction on the system. It's not as simple as changing a line of code. It has to be found, fixed, tested, reviewed to ensure nothing else breaks in the system. Unless you have dealt with this on a daily basis, please don't assume it's so easy. And, if you have dealt with it on a daily basis (AKA, you work in the software industry) you should know that this is a hard thing to do.