Is something wrong with the large, massive sheet?
And don't forget, for almost any conceivable gravitational field and any given level of sensitivity, there is a calculable scale below which the field is indistinguishable from a uniform field. That means that this problem is not irrelevant to the laboratory.
Or am I wrong?
You're missing the point. You might as relevantly have pointed out that the situation where someone is on a rocketship without knowing it is unlikely ever to arise. The equivalence principle is a far-from-obvious statement about the physical nature of gravity and inertia, not merely a practical limitation on figuring out whether you're travelling on a rocketship. Instead of pettifogging about the practical details of a given example, why don't you try to focus on the principle that the example was meant to illustrate?
The mass of the Sun 'attracts' the mass of our planet ... not directly, but by creating a field in which the motion of our planet is effected. Is that correct?
If that is a correct notion, the question then becomes, what about mass causes this 'field' influencing other masses? ... Don't physicists define the 'thing' causing the influence, creating the field, 'gravitons'? And if there are gravitons, doesn't it appear that these gravitons are actually influencing the spacetime of the universe (the background field in which masses exist), and that affected spacetime is that which then acts upon the other mass(es)?