Actually, Cloud usually does manage to protect the people he cares about. However, he fails on three significant occassions
1. After the death of Tifa's mother, when he and Tifa fall off the bridge and he is blamed for her inuries.
2. When Sephiroth kills his mother in Nibelheim.
3. When Sephiroth kills Aerith.
It appears to be after the first incident that he become even more withdrawn and solitary, and at some point forms the resolution to become a powerful warrior, 'the best of the best', like Sephiroth. Psychologically, his motives seem obvious, even if he himself is unaware of them.
The other two crucial losses are (ironically, as if to rub in his failure) inflicted by the very man he idolised).
He fails to become a SOLDIER, and his mother dies; he then kills Sephiroth, which is something only a First Class SOLDIER ought to be able to do. When he finally emerges from his mako coma, he's convinced that he is a First Class SOLDIER. Again, what's going on in his subconscious seems fairly obvious, though his delusion must make it even harder for him to explain to himself why he couldn't save his mother.
However, he isn't a real SOLDIER, and Aerith dies.
If a part of your subconscious believes that you're not capable of protecting the people you care about, the obvious solution is to not care about people. Hence Cloud's gruff and cold persona, which in reality is the thinnest of shells. This is why his subconscious understands that Tifa poses a threat to that protective persona in several ways. She has memories which contradict the ones his subconscious has fabricated, and he really, deeply cares about her.