Do objects with mass "suck in" spacetime?

You can indeed think of gravity as changing which direction is "forward" for a moving object. But the key idea is that gravity acts in spacetime, not just space.

An object that is sitting still in space is still "moving" in spacetime*, it's just that all its "motion" is through time, not through space. In other words, the forward direction for this particle is time. Gravity changes the forward direction for the particle to include a little bit of space as well, so it starts moving through space. That's how gravity can accelerate a particle from rest.

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^{*Technical note: "motion" through spacetime means the transition between different points along a worldline as the value of the variable parametrizing the worldline (proper time) changes. Not to be confused with motion through space, which is the transition between different points along a spatial path as the value of the variable parametrizing that path (coordinate time) changes.}