It seems to me that efficient treatment of a loop-the-loop problem will always feature both assessment in terms of force and acceleration, and assessment in terms of energy.
Without putting pen to paper: here is how I think about it:
First calculate how much velocity is needed at the point where the vehicle is at the top of the loop. Given the radius at the top the velocity much be such that the vehicle is pulling a higher G-load than the Earth's gravity.
For a vehicle without propulsion of its own:
The amount of velocity that it loses in climbing from the entry of the loop to the top of the loop can be calculated in terms of change of kinetic energy. The sum of potential energy and kinetic energy is a conserved quantity, so from the height difference you can infer the loss of velocity.
In order to make it around the loop-the-loop the velocity of the vehicle at it enters the loop must be sufficient so as to still have sufficient velocity as it reaches the top of the loop.
Of the cuff: can the velocity change be tracked purely in terms of force and acceleration?
Take the case of a vehicle that is coasting up a ramp. We take the idealized case of frictionless motion. As the vehicle moves up the ramp it is slowing down. As the vehicle is coasting up the ramp, at what height above the starting height does it come to a standstill?
Interestingly, that height difference is not dependent on how steep the ramp is. On a very shallow ramp the component of gravity slowing the vehicle down is small, so the vehicle will coast a comparatively long distance. On a steep ramp the component of gravity slowing the vehicle down is large, with a correspondingly shorter distance of moving along the ramp.
No matter what the angle of the ramp is, the combination of coasted distance along the ramp and angle of the ramp has the vehicle always coming to a standstill at the same height. (Of course, implicitly that is gravitational potential energy: product of gravitational acceleration and height difference.)
Given a specific angle of the ramp you can always work out the time it takes from starting the ramp to coming to a standstill. Then with the acceleration and time interval known you can arrive at velocity difference ( $v = v_0 + a \cdot t$).
A loop-the-loop is a ramp too, but instead of being a ramp with a uniform angle to the horizontal it has an ever increasing angle.
But it gets complicated.
As the vehicle ascends the loop two forces must be taken into account: the centripetal force exerted by the loop on the vehicle, and the gravitational acceleration that the vehicle is subject to.
The wonderful thing about expressing a problem in terms of interconversion of potential and kinetic energy is that it isn't necessary to keep track of the direction of the acceleration.
As you ascend, no matter in what direction that ascent is, you lose velocity, with the loss of velocity such that the sum of potential energy and kinetic energy is conserved.
(In the specific case of a loop-the-loop: for the purpose of the calculation the motion can be thought of as 1-dimensional motion. The location of the vehicle along the loop gives the direction of its velocity.)
The expression of interconversion of potential energy and kinetic energy is the work-energy theorem. For a derivation: a 2023 answer by me with derivation of the work-energy theorem
