To trust or not to trust.....very good question.
I can only speak for myself and maybe it'll help.
Trust in a relationship is TOP priority for the foundation as with respect for one another.
As far as your BF " not trustworthy" as for you there had to be "something" that made you feel to look at his phone.
Don't blame yourself for looking . Most couples whether they admit it or not look at their partners phone , usually 9 out of 10 do. I personally do not lock or hide mine at all if asked I'll gladly give it to my partner. As will I expect the same in return especially if there is possible matrimony involved. It is also said that if spouses traded phones for a day , that at the end of that day most would be asking questions about what they've seen.
Privacy .......is there and should there be privacy in a relationship? If there is a foundation of mutual trust and respect my answer is " no ".
How to address the messages you've found ....easy ....ask about them .....hard ....what to do with and take his response....that has to totally your decision ( not an easy one). Hopefully he'll come clean and rebuild your trust, or deny , lie and cover his tracks better, or put it back on you for violating his " privacy" . If it's not the first suggestion of rebuilding .....then unfortunately the relationship you have with him is doomed. Simple as that. Only you know what you feel and all the support or advice means nothing . I do hope that you and he find a mutual solution whether to stay or go. Your situation is terrible and I certainly feel what you are going through. Good luck to you him.