It's not just about the base algorithm. It's also about the memory needed to run it, and the clockspeed. For example, even the hardest problem you can imagine, if it has a verifier algorithm that fits in 4k (which means the solution itself can be much larger than 4k), then you can simply do a basic brute force search over the solution space. That doesn't mean this algorithm is very intelligent; it's only very capable if you have a sufficiently fast computer; although indeed brute force is only feasible for the simplest tasks in practice, so the idea that algorithms (of increasing sizes) enable (greater) intelligence is definitely a part of the story, but not the whole story. You can also think of DNA, which represents a recipe for our bodies and brain, which we then use (essentially as an "algorithm") to learn things, with degrees of freedom (memory) far surpassing what DNA stores.
Now if you had a very good chess program running in very constrained (dynamic/RAM) memory, then that'd be partially more revealing. From a cursory search there's a 1800 ELO engine for the C64, which seems very impressive but very far from the best human players.
I'd be interested to see a curve of ELO x Avaliable RAM for the best chess engines (up to given RAM), and how that compares to other games and activities.
On RAM vs ROM (program size) memory, I think at a high level dynamic memory helps you keep track of search paths in a large tree search, saving you some computation. Program size tends to enable improving the effectiveness of your search heuristic, as well as pre-computing e.g. initial and final game optimal moves (potentially saving arbitrarily much compute). I like thinking about those things because I think the search paradigm is pretty informative of computation (and even intelligence) in general. Almost every problem is basically some kind of heuristic search in some kind of space. And you tend to get better at things by refining your heuristics (usually through some experimental training process or theoretical insight), considering more options, exploring deeper consequences, etc..
I think what really defines humans isn't really our ability to solve problems or play chess well etc. (although that's extremely useful and also enjoyable most of the time), it's really our emotions and inner world. We are not really Thinking Machines in essence, we're Feeling Machines most significantly. The thinking part is a neat instrumental part :) We can delegate thinking to machines but what we cannot extinguish is feeling or the human "soul", because that is the source of all meaning.
