Well, gravity is a force and it isn't. What is a force anyway? It's what makes you accelerate, which is already a statement about a second-order derivative of one variable with respect to another, and now all of a sudden your coordinate system is important.
The point being made when someone says "gravity isn't a force" is that, if you express a body's location in spacetime, not space as a function of proper, not "ordinary" time along its path, gravity doesn't appear in the resulting generalization of Newton's second law in the same way as other forces do. In that coordinate system, the equation can be written as $\color{blue}{\ddot{x}^\mu}+\color{red}{\Gamma^\mu_{\nu\rho}\dot{x}^\nu\dot{x}^\rho}-\color{limegreen}{a^\mu}=0$, where the red (green) part is gravity (other forces). But this red/green distinction looks different, or disappears, if you look at things another, mathematically equivalent way. In particular:
- Putting on Newton's hat This is the less elegant of two options I'll mention, one that uses pre-relativistic coordinates. If you look at the body's location in space, not spacetime as a function of ordinary, not proper time, the red term looks like the green term, hence like the stuff you learned from Newton. In particular, $\frac{dp^i}{dt}\ne0$.
- Putting on Einstein's hat Even more elegantly, we don't need to leave behind the coordinates I suggested first to change our perspective. As @jawheele notes in a comment, we unlock the real power of GR if we use a covariant derivative as per the no-red formulation $\color{blue}{\dot{x}^\nu\nabla_\nu\dot{x}^\mu}-\color{limegreen}{a^\mu}=0$. This time, the equation's terms manifestly transform as a tensor, making the blue term the unique simplest coordinate-invariant notion of acceleration.
The main advantage of the $\Gamma$-based version is doing calculations we can relate back to familiar coordinates. This not only recovers Newtonian gravity in a suitable limit, it computes a correction to it.
Regarding the first bullet point above, have you ever spun on a big wheel? There's a similar perspective-changing procedure that says the dizziness you're feeling is due to something that's "not a force". You're still dizzy, though. This isn't a contradiction; they're just two different ways of deciding what counts as a force.
The good news is we don't need to "forget" GR to understand a gravity assist. How does it work? It exploits the fact that, if a planet's in the right place at the right time for you, the red term is very different from what the Sun alone would normally give you there. This has implications for the blue part even without wasting fuel on the green part. Or you can explain it without GR; your choice.
