Why is the situation of 'accelerating frame in flat spacetime' not equivalent to 'staying stationary in a constant gravitational field'? One popular form of the equivalence principle states that the effects of a gravitational field in a small enough region of spacetime are indistinguishable from those of being in a uniformly accelerating frame. I interpret this statement as whenever we have a constant gravitational field, we can replace that situation with a uniformly accelerating frame.
Why is a uniformly accelerated frame not replaceable by a constant gravitational field? It appears to me that any constant acceleration can be mimicked by or replaced an equivalent situation with a suitable constant gravitational field. Therefore, even if we have an observer accelerating in flat spacetime, I can replace that scenario by a gravitational field. Let me give an example to illustrate what I mean.
Suppose an airplane is flying horizontally with an acceleration forward where spacetime is flat. If a traveler is not allowed to look outside the airplane, how will he decide whether the push that he feels on his back against his seat is due to genuine acceleration or a gravitational field?
 A: 
I interpret this statement as whenever we have a constant gravitational field, I can replace that situation with a uniformly accelerating frame.

This is almost OK, but you really need to mention "locally" in this statement. If you have a constant gravitational field you can locally replace that situation with a uniformly accelerating frame. The "locally" is an essential restriction because it excludes situations with detectable tidal effects.

Why a uniformly accelerated frame is not replaceable by a constant gravitational field?

It is, again with the restriction "locally".

Suppose an aeroplane is flying horizontally with an acceleration forward where spacetime is flat. If a traveller is not allowed to look outside the aeroplane, how will he decide whether the push that he feels on his back against his seat is due to genuine acceleration or a gravitational field?

You never feel uniform gravitation. Ever. The only gravitation that you can feel is tidal gravity, and that is excluded by the "locally" restriction.
What you feel is always proper acceleration, and that is caused exclusively by real forces not fictitious forces (including gravity). So regardless of whether the traveler is using an accelerated frame or not the force that they feel is due to the contact forces between the seat and the body. It is not due to either gravity or other fictitious forces.
As to how he recognizes if he is using an accelerating frame or not, that is simple. If his coordinate acceleration is zero while his proper acceleration is non-zero then his frame is accelerating. If it is non-inertial then there is no possibility to locally detect if the fictitious force is gravity or some other fictitious force (they are equivalent).
