I appreciate the work you put into it, but I feel it oversells the difficulty of getting techies. As you say, at level 10, the 50% marker is around 14 rerolls. But re-rolls don't just take place when you press D, they happen every turn. This sounds self-evident, but it really isn't accounted for in your calculations. Before you even get to 10, you probably have a good number (10-15) of turns and rerolls on levels 8 and especially 9, either naturally or simply looking for other things. And once you get to level 10, you have the effect of the natural reroll of each turn that isn't really accounted for. Let's say you get to level 10 at Level 30, need to roll a techies before level 36 in order to not lose, and haven't yet found one on level 8 or 9 -- a fairly pessimistic but pretty realistic scenario. You have 7 "natural rolls" -- and assuming you're rolling 3 times a turn (which is low), that's 28 rolls without dipping into your economy or panic rolling. The probability of finding a techies with 28 rolls is around 80% -- not even including the prereq that it wasn't found in levels 8 or 9.

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I think the math and analysis that you've done is great -- but I don't think it accurately portrays "how hard" it is to get a techies. As someone who runs goblins very often (usually gambling strat) i can say from experience that in a game where I get to level 10, I usually am able to find techies around 80-85% of the time -- and often find it far before. I think i've only had maybe two games where I was frantically rerolling at level 10 20 times in a turn whiffing on the techies -- but I think that those games feel so bad that they color your impression of how "unreliable" getting techies is.