PS. Of course you should rush after the lone worker/settler with everything you've got. You never know - it
might have a move of two over rough terrain and get away!
I didn't say "with everything you got"

. I said with a troop. Preferably an infantry so it can defend itself and\or the slave(s). You might not bring enough troops to attack. But I bring excess troops in the hundreds

.
I really don't want to argue pointlessly but I just want to prove my point. Different strategies work for different people. And since I bring
hundreds of troops into battle I can use a few infantry to capture some slaves. It is not as if like you said I "use everything I got"

.
And no I don't go bankrupt from all the troops. I trade techs with the AI and usually make a thousand gold per turn. I usually end up with 5,000-20,000+ Gold and get about 50-1200+ gold per turn. But my military isn't huge. Only about 500-600 troops.
But because I don't attack all the enemies at a time, I bring excess troops of hundreds like I said. If a few troops capturing slaves hold me back from conquering my objectives then I'm obviously not building enough troops. But that never occured to me and never will because I know I'm building enough troops.
I'm not saying you can't do this you can't do that but if a few troops hold you back then you obviously don't have enough.
(edit) And I'm not saying I'm a really good civ player. And i'm not saying I'm better than you or anyone else.