The main reason I think reproduction wouldn't work for sperm is because they have no way of obtaining energy.
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Also, for sperm to be able reproduce and still reach the egg cells, they would have to simultaneously gain the ability to gather energy.
I made a league bot (named Nanite Detonators) which does not do any energy gathering whatsoever, no shooting, no tie feeding, nothing. Starting with 5 bots, it reproduces and flies all over the place, killing any enemy bots it notices (with an instakill tie attack - setting strvenom or strpoison to 10000 or so). It reaches a max population of around 550 bots, if the opponents aren't particularly tough (after which they begin to die off, but when you've got 550 bots flying around the map at high speed, pretty much nothing can hide). In a sim with wrapping sides, it wipes out every bot I tried it against from any league. When I made another bot look friendly to it, that bot still lost due to ( a ) trying to attack or feed on the nanites and getting killed in retaliation, and ( b ) there were no veggies to eat due to the nanites killing veggies shortly after they (re)spawned.
Given a bot starting with 3000 energy and 1000 body, which is what a league bot starts with, that single bot should be quite capable of splitting off half its energy and body to create 50 or more sperm bots, which would have quite a good chance of finding any eggs out there. If the main bot waited until it had 30000 energy and 10000 body to split off half of it, and that half went towards creating nothing but sperm, then it would make an equivalent amount of sperm bots (550) as the previously mentioned group of 5 bots using all their energy and body. (Here's a bit of very simple math showing why: If 5 bots with 3000 energy and 1000 body can make 550 sperm by using all their energy and body, then 1 bot using half of its could make 55, and if 1 bot had 30000 energy and 10000 body, and still spent half of it, it would be spending 10x as much resources as the 1 bot using half of 3000e/1000b. So, it could make 10 * 55 bots, which would be 550.)
Note: I didn't read the entire thread, I mostly just read the thread-necromancer's post.