I don't strain my wort going into the fermenter, and for over 300 batches it's worked well!
You also don't strain if you're going to a secondary. You will be siphoning ("racking") quietly to avoid any splashing or oxidation, with the tip of the siphon at the very bottom of the receiving vessel to avoid any changes of oxygen pick up.
I don't use a clearing vessel ("secondary") very often, and it's not needed, but when I do I found that just careful siphoning from the fermenter avoids any trub being transferred.
The heavier particles like proteins, hops debris, and even spent yeast will all fall to the bottom of the fermenter. If given a chance to sit for at least a few days after fermentation ends, this will form a pretty compact trub layer and you just siphon from above that into the bottling bucket or a clearing carboy if using one. If you ARE using a clearing vessel, it should be a carboy and not a bucket. A bucket has a very wide headspace, inappropriate for a clearing vessel.
It sounds like it's a bit backwards. The bucket is for primary, for fermentation, and the carboy is a clearing vessel.