Is energy more fundamental than force? I was wondering, there are several examples where energy comes in very handy. Many problems in kinematics which are really challenging if you could use only forces, can be solved effortlessly by converting into terms of energy.

(Illustrative image)

What makes most force calculations tedious and energy calculations dead simple? 
Intuitively, is it right to think of energy as a more fundamental quantity than force?
 A: No, it is false. You have this impression just because you are considering one-dimensional problems where the forces are conservative or they do not produce work. Newton's equation of motion, in this situation,  is essentially equivalent to the theorem of mechanical energy conservation.  If there are non-conservative forces and/or the motion involves more than one dimension, then the theorem of energy conservation may give some information, but it is by no means sufficient to get the motion of the system.
A: Neither energy nor forces are something we know to exist. And neither of them are really "fundamental".
We often want to predict what will happen when we release a roller-coaster cart down a track, or throw a satellite at Jupiter, or any of a number of other things we might do. Presumably we cannot ever know how Mother Nature truly functions (if that even makes sense to ask about), but we have, over the centuries, noticed a few striking patterns.
In order to exploit those patterns so that we can make predictions, we have invented some seemingly helpful quantities and given them names. We have then set up mathematical relations that we believe these quantities follow (or at least are close to following).
It turns out that for any given situation, there can be several of these helpful quantities and corresponding mathematical relations that one can use. One common way is to use forces, and the mathematical relationship that is Newton's second law. Another way is to use energies and the mathematical relationship that is the principle of least action. It depends entirely on the situation (both the setup and the actual answer you're after) which one is easier to work with, but neither is more fundamental than the other.
In the case of your specific setup, assuming there is no friction and that the gravitational field is uniform, working with energies is far easier than working with forces, because one of the patterns we have noticed is that this "energy" quantity is conserved. But the energy approach is so much easier only because you're asking about the speed of the cart (energy is much more closely linked to speed than forces are, which also plays a part in why energy considerations makes it easier to answer questions about speed). If you had instead asked for the time it took to get to the bottom, then suddenly the difference in effort between the two approaches isn't that large any more.
A: Let me discuss it for classical mechanics
While Work-Energy mechanics is very powerful, it also has an achilles heel.
A necessary condition for Work-Energy mechanics is that the system is such that an unambiguous potential energy can be defined. 
You'd think that you would have to invent a very contrived case to run into that problem. But here is the example given in many textbooks: a ball rolling frictionless on an inclined plate. When you move that ball around, and you return it to the same location as when you started, that ball won't be in the same orientation as when you started. How the orientation of that ball comes out depends on details of how it was moved around. 
There is no unambiguous integral for that case, so Work-Energy mechanics cannot solve that case exhaustively.  
It's kind of a geometric problem, it tends to be about orientation in one form or another. You need a well-defined integral for the potential energy.
When Work-Energy mechanics cannot solve the case exhaustively you fall back on Force mechanics. 
I guess it should be thought of as a trade-off.
Work-Energy mechanics is like a fast car. You get the speed, but there are places where it cannot go. In this analogy Force mechanics is like a tank. It's a lumbering piece of kit, but there is no place where it cannot go.
A: 
Intuitively, is it right to think of Energy as a more fundamental quantity than Force?

The “fundamental-ness” of something is difficult to quantify so it is hard to say whether something is more or less fundamental. However, I would tend to agree with you, but for different reasons. 
There are two basic approaches to classical mechanics. The Newtonian approach is based on forces, and the Lagrangian/Hamiltonian approach is based on energy. In quantum mechanics only the Lagrangian/Hamiltonian approach is used, the Newtonian approach doesn’t apply, and forces can be difficult to define let alone calculate. So insofar as QM is considered more fundamental than Classical Mechanics I think you would consider energy more fundamental than force. 
One other point that may lead to the idea of energy being more fundamental than force is that energy is conserved while force is not. Again, fundamental-ness is fairly vague so it is not clear that conservation is a criterion, but it does seem reasonable to consider it briefly. 
You mention simplicity, which is not as clear cut as these other points. Some (few) problems are simpler in terms of forces, especially ones with friction. It is good to know forces also for those problems. You can apply Lagrangian methods to dissipative systems, but it is not simple. I think your overall impression is correct, that there are more problems which are simpler in the Lagrangian approach than are simpler in the Newtonian approach. 
