I guess I’m thinking of Centripetal force but I’m not sure you would feel that either without a physical connection to the orbital center.
I’ve read a lot of Robert Forward who wrote extensively on tether assisted maneuvering which creates a physical connection leading to a felt force.
If the orbiting object came into Earth’s gravity well too shallow, it would be slung in the wrong direction away from the planet. If it came in too steep, it would miss the gravity well altogether or have to come back for a second assist later. Fortunately, these calculations are solid formulae tested and confirmed many, many times.
I remember studying the use of gravity assists in the astrophysics part of my physics courses in college. While they used to do the calculations on paper with slide rules, computers can do it all now.
It’s the same sort of thing for which they had to account when sending our boys to the moon. Come in too shallow and they crash into the lunar surface; too steep and they miss the moon’s gravity and careen into space.
As far as “feeling” it, the properties of force in space and force on Earth are different due to the constancy of gravity at the Earth’s surface. We can calculate the acceleration of gravity on Earth as 9.6 m/s^2. Since there’s no atmosphere in space, there’s nothing to hinder the rate of acceleration in space up to the force of thrust. When you hit a gravity well, your trajectory changes slightly, but if you hit it just right, you reach escape velocity of that object’s gravity and gain speed from the ejection, much the same way a NASCAR driver speeds up at the peak of a turn to gain momentum on the downslope.