"What force pushes a glider to fly?"
It depends on what reference frame we are using.
Newton's first law of motion is Force = Mass times Acceleration. In linear straight-line flight, acceleration is zero, so net force is zero. No net force exists in linear (straight-line) gliding flight.
We should then consider the exact meaning of the words "What force pushes a glider to fly?"
These words are basically asking what are all the forces that exist that exert a component parallel to the flight path, acting in the generally forward rather than the generally backward direction. (Note that the question did not use the word "thrust", which has a different, specific, well-defined meaning in aviation. "Thrust" is zero in gliding flight.)
The answer to the question depends on precisely what we mean by "flight path".
If we are looking at the flight path through the airmass, there is no such aerodynamic force, but gravity does exert a force component that acts against the direction of the drag vector, i.e. parallel to the flight path and acting in the generally forward direction, even though gravity is purely vertical. So there's one answer.
If we are looking at the flight path relative to the ground, the answer gets more complicated, and depends on whether the glider is descending, climbing, or maintaining constant altitude. Consider the case where a glider is maintaining exactly constant altitude in slope lift. Now what force is providing a component acting parallel to the trajectory, in the generally forward direction? Not gravity. And while the net aerodynamic force acts straight up vertically, the component of the net aerodynamic force that we call the lift vector does contain a component that acts parallel to the trajectory in the generally forward direction, and thus opposes the component of the drag vector that acts parallel to the trajectory in the generally backwards direction. But take care not to confuse this statement with a claim that lift is actually helping to oppose drag-- that's not the case. Lift and drag are orthogonal (i.e. are perpendicular to each other).
The argument immediately above may strike some readers as an arcane way of playing "games" with components of vectors. But in truth the same could be said of the claim that gravity is helping to "push" a glider through the air in some way. The key point is that in straight-line gliding flight, lift, drag, and weight form a closed vector triangle, with zero net force. Which components of this triangle may be considered to be contributing a component that "pushes" forward along the trajectory, depends upon which reference frame we are viewing the trajectory from.
The analysis gets even weirder if we are looking at the trajectory relative to the ground, and the glider is climbing. There are even valid reference frames where the glider is moving backwards. Now what is the direction of the force component that we would consider to be purely a "pushing" force?
For example, consider a glider that is slowly ascending straight up relative to the ground in powerful mountain wave lift. This happens often. Since the lift and drag vector are always defined relative to the airmass rather than relative to the ground, for a given steady-state airspeed they retain the same orientation in space for a given angle-of-attack of the wing, just as the glider retains the same pitch attitude in space, no matter which way the airmass is moving relative to the ground. As the glider rises slowly straight up, it's obvious that the lift vector and the drag vector now both contain components that act in the direction of the trajectory relative to the ground, while the weight vector does not. Similarly, it's not hard to imagine a case where the glider is drifting backwards and climbing along such a trajectory that only the drag vector contains any component that acts along the directory of the trajectory relative to the ground.
The idea that a component of the lift vector is helping to pull the glider forward along the aircraft's trajectory as viewed from the ground is only true when the aircraft's achieved glide ratio is better than the L/D ratio, or when the aircraft is climbing (unless the aircraft is drifting backwards along a climb path that is flatter than the direction of the drag vector). If there is no tailwind, this means that the air must be rising.
It's usually most useful to focus on the glider's flight path through the airmass rather than the glider's trajectory relative to the ground, or relative to other reference frames, but all of these points of view are technically valid.
When we are using the glider's flight path through the airmass as our reference frame, the answer to the question ""What force pushes a glider to fly?" is "the component of the weight or gravity vector which acts parallel to the flight path." In this reference frame, neither lift nor drag are exerting any "pushing" force, i.e. any force acting parallel to the flight path and pointing generally forward rather than backwards.
Note that a related question may be asked-- "what powers a glider in flight"? Work is force times distance, and power is work per time. Again, the answer will depend on whether we are looking at the work done along the direction of the trajectory relative to the airmass, or the direction of the trajectory relative the ground. In the former case the answer is simply "the component of the weight vector that acts parallel to the flight path", while in the latter case the answer depends on the direction that the airmass is moving relative to the ground.
Related ASE questions and answers (including some useful diagrams)
What produces Thrust along the line of flight in a glider? (Question)
What produces Thrust along the line of flight in a glider? (Answer)
'Gravitational' power vs. engine power (Question)
'Gravitational' power vs. engine power (Answer)