This is not a question about GoL directly but more generally about CA. Do you have a sense for what the probability of a random CA is to be TME? Do you have any idea of how to automate the process, if not in general, then at least for a class of CA?
This is a very good question, but I'm not sure I can give a very good answer.
I'm not sure the "probability" part of the question is even well-defined, let alone answerable, unless you state a specific rulespace -- two-state range-1 Moore-neighborhood CAs on a square lattice, or three-state range-2 isotropic CAs on a hexagonal lattice, or what have you.
Basically, you just have to be able to demonstrate a working universal logic gate (a NAND gate or a NOR gate) in a candidate rule, and you've pretty much got Turing-machine equivalence.
The problem is, a lot more rules are computationally universal than you'd think when you first look at them. This is because it's often possible to get a candidate rule to act like a completely different rule, by filling the universe with something other than empty space.
So you can't just try out a few random-soup patterns, dash off a quick proof that "this rule necessarily explodes uncontrollably in all directions, so it's impossible for any circuitry to survive" or anything along those lines. What if you start with a universe of all ON cells, or a checkerboard of ON and OFF?
There are lots of rules where signals can propagate beautifully through that kind of non-empty medium, and occasionally some kind of Turing-complete mechanism might be found there, along the lines of what Matthew Cook did with Rule 110. So you really have to look at a lot of options before you can say for sure that a rule does not support universal computation -- and so far, it seems like a very tricky problem to automate the process of looking.
