If I tuck my elbows, that would make it just a shoulder/tricept movement. My chest would be completely disengaged. Is that good? How would i work out my chest, then? I know of other exercises to work my tricepts if I choose to isolate them. I also don't see how your press would be easier on the shoulders. Maybe it would be more natural, but for they would be used a lot more.
Nope, tucking your elbows still makes it a chest exercise. It's a trade-off though. 90 degree elbows hit your chest a bit more, but are hell on your shoulders. The wider your grip, the more chest is involved.
If you want to lift chest with almost no risk of injury to your shoulders, do DB bench.