I use first person pov often. It's an effective way to literally "get into" your character's head and can establish a connection between the reader and protagonist right from the start. You don't have to use tags like, "he thought" or, "she mused" when reflecting on an idea or event (though you could certainly avoid those in third person via clever technique). The biggest drawback I've encountered with first person pov has been maintaining mystery and suspense on the part of the protagonist. It's difficult to justify a main character making any surprise moves when we've been living inside their head for 200+ pages. I've changed more than a few short stories to third person because of this.
You could certainly write a story from the pov of a journaling character, though you'd probably want to start by establishing a frame of distance between the character and their experience. In other words, how much time has passed between the event and when they sat down to journal? How much do they really remember and is their memory accurate? Most people don't remember things with 100% accuracy; you could have some fun here. You'd also want to think about the level of detail afforded by this type of telling. They're recollecting events, not living them, so, for example, if the protagonist is writing about an argument he had with his girlfriend, he probably wouldn't recall her sighing or looking down, but he might mention that she was biting her nails or stomping her feet, i.e. big, obvious, memorable things that allow you to "show" the scene playing out without being there in the moment.
Personally, I'm not a big fan of the "tales from my diary" perspective, but that's just my own bias. If it's something that interests you, then why not try your hand at some experimental free writes/flash fiction?