This might sound like a lot of work, but you could undertake a study of world religions and/or philosophy with her. You could also buy her some decent beginning philosophy books-- there are some written for children, some written with an encyclopedic overview, and some written as short biographical studies of famous philosophers. Maybe that or similar materials would interest her and answer some questions-- not about what life means, but about the types of questions others have asked along the way on the topic, so she wouldn't feel so alone in her journey.
This is what I did with my junior philosopher at this age. It was very helpful to her. She does not seem to have a spiritual sense of need for a deity, however, and this has troubled her somewhat throughout her life. She can see how fulfilled/comforted religious observance makes her friends and some extended family... but she doesn't seem to have what the Catholics refer to as "the gift of faith." It was quite painful to her that more devout family members clearly wanted her to lie to them for social reasons on this particular score. That kind of existential conflict is quite difficult for someone who truly feels NOTHING spiritually. Just noting that this may be the case here, since you mention that her educational environment is a religious one. This might explain her preoccupation with social lies (or inaccuracies, really).
She did develop a fascination with world faiths, mythos, and world history as a result of that early exposure, so it was good in any case.
Indicentally, my DD is still deeply philosophical and empathetic, but she has grown to be a hard-nosed pragmatist with a blazing sense of social justice, too.
Does your 4yo feel that you are "sheltering" her, or that others are treating her as a child too much for her tastes?
How does she handle what we tend to think of as "adult" topics? That is, some of the more distressing aspects of human history? My daughter was very much like yours at 3 and 4. She was hungry for... well,
reality. She was very much ready to deal with the complexities, even though we have had to remind ourselves that she was a child who lacked a certain amount of life experience necessary for understanding those things the way that adults do/did. Her sophistication and natural understanding was and is a little freakish. Honestly, I used to think that the stories about the Dalai Lama were hogwash, but having lived with my DD... I have seen that eerie kind of "knowing" vibe, and it does feel very supernatural sometimes.
It's also possible that she is (at least somewhat) manipulating you, if claims of boredom (or expressing these admittedly very dark thoughts) get you to respond in a more overtly engaged and alert manner. Frankly, a PG child who is loquacious can be an e-ticket and by the end of the day, parental fatigue is a real problem. My DD went through a phase of saying-- well, pretty much ANYTHING outrageous enough to elicit an animated or interesting response from us. That might explain why she claims "boredom" when as far as you can tell, nothing could be further from the truth. It's particularly true that if she's LOOKING to convince you to let her expand her activity schedule, claims of boredom would seem like a great lever to use.
Some of it, as others note, may be asynchrony, pure and simple. It's a frustrating thing to feel like you're imprisoned in the body and life of a preschooler when you can see the world with the intellectual sophistication of a much older person. Without the emotional maturity (or natural inclination) to process that frustration well, or to temper it with the knowledge that childhood is relatively short, and will be over soon, or that adulthood has many benefits to balance the responsibilities... well, that's not a good mixture. Train wreck is right.
