You don't have to trust "the system" with Bitcoin.  The algorithms and crypto are public.  As long as a majority of hashpower is operated by honest nodes, the blockchain is the _truth_ of who owns which coins. (I have a concern about how we can _know_ that the majority of hashpower is operated by honest nodes, but that's another question altogether).

You don't have to trust the Bitcoin "system", but you have to trust people. You have to trust vendors whom you pay to give you what you paid for (this is true even in a cash economy), but vendors who repeatedly rip off their customers tend to go out of business. You have to trust people with whom you share your private keys, such as family and friends in the event of your death (though secret-sharing protocols can keep them from knowing the secret without a quorum). Note the pattern here: wetware has to be trusted (sometimes), but the algorithms don't have to be trusted.
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Fran