Whenever you group together, you lose out on collective gold and xp. Grouping should only be done if something offsets this gold and xp loss, such as killing an enemy or securing an objective. Otherwise, if you're grouping for the sake of grouping, you're losing out.
Taking the tower early frees up the enemy top laner to roam a bit more, since the next tower takes a shorter time to get to with a recall if you intend to push. If you freeze the lane, you provide no pressure on the enemy laner to defend it, freeing him up to secure objectives. The global gold is nice for your team, but no pressure is no pressure.
Since this streamer wanted to farm, I can infer that an early tower would either force him to push lane (and fall farther ahead if he gets ganked or caught out) or freeze lane (which would provide no pressure on the enemy). Not taking tower would allow him to farm safely while still providing pressure at the enemy tower. Now you said trade, so I assumed he didn't die because you didn't clarify that. If he can farm better than enemy, it doesn't matter if he lost the early trades if his gold income is higher.