This is an interesting way of framing the idea, but it's not a question of traveling in DNA space from some point (eukaryotic cell) to another specific point (m

This is an interesting way of framing the idea, but it's not a question of traveling in DNA space from some point (eukaryotic cell) to another specific point (mushroom): that would be very difficult in the way that you're talking about. Imagine flipping a fair coin 256 times. The particular outcome ('HTTTTHHTTTTTTTTHTHHHTHHTHTHHHHH...') is extremely difficult to replicate, but getting any outcome is very easy: just flip the coins again. In this case we also have a lot of selection bias: all the paths through DNA space that don't result in intelligent life don't result in anyone having this conversation. Regarding the curse of dimensionality: it's a statement about the available data rapidly becoming sparse in high dimensional spaces. It doesn't really say that high dimensional spaces are necessarily sparse, it's just hard to "fill" them in with the amount of data available.