...would the flipline save me (or would I simply fall down the tree)...?
If you follow your instincts and try to "hug the tree", you're in for a painful lesson. You need to have a plan and teach yourself how to react correctly. The lanyard, to work, needs forces in two directions...
down, and
out (away from the tree)... the
down is easy, since gravity is happy to help you with that one. It keeps your gaffs properly sunk into the tree, stopping downward travel enough that the lanyard or flipline can stop the really heavy part of you from having to wait for that sudden stop at the end of a long fall.
The
out part requires input from you. It shifts the downward force of your weight onto the lanyard and gaff and tree triangle... the holy triangle of climbing poles and pole-like structures like trees... and
YOU are the hypotenuse of what is nearly a right triangle. That lower angle formed by you and the trunk of the tree is critical... you need to find the correct one for climbing and work positioning that is right for you and the tree you're on.
When you gaff out and panic, you will likely either do nothing constructive or try to hug the tree, which turns you and the tree into two parallel lines, and the triangle disappears. Then gravity tries to send you and your lanyard to the sudden stop at the bottom of the tree. You need to keep the triangle, because even if it hurts a little, it will stop you.
On a tree (it's a little different on a telephone pole or dead spar with no bark) you need to take advantage of other instincts you have... clench your fists (an instinct in scary situations), rotate your fists outward so the back of your hands and forearms are toward the tree, and form a 90 degree bend at the elbow (a semi-fetal positioning of the arms, also instinctive) and push outward. The skin on the inside of your arms is nowhere near as tough as on the outside, and the major veins and arteries are right underneath of there. Splinters can do serious, even life-threatening damage.
This action will put tension back on the lanyard, and in combination with forcing yourself to reform the triangle immediately with your feet and legs or even your crotch, allows the lanyard to grab the opposite side of the tree. The friction created, in combination with the friction created by your legs against the tree, will stop you almost immediately. But, it's something that you need to try and practice a few times until it sinks into your brain that it will save you.
Easiest way to do that is to put on a leather jacket and try it on a few trees of various sizes... at about three feet off the ground. On poles, and trees smaller than about 15 inches, you may only have to push away from the tree with your hands and squeeze the tree with your legs at the same time. The important thing is leaning back and pushing back to reform the triangle.