I will take the great examples of the well-known objects in the Universe to demonstrate an example to answer your question: The Sun and Jupiter.
However, as the core became denser and denser, it reached temperatures
at which fusion could take place, but it did not become dense enough
for the matter to become electron degenerate, right? Only later in its
life, when the star's hydrogen ran out, would it become dense enough
That is because, despite the fact that the Sun's core is nearly a tenth of a tonne per m3 dense (150 g/cm3), the Sun's core is still balanced outwards by thermal pressure. The Sun fuses hydrogen to helium, and that liberates a lot of energy equivalent to nearly a trillion H-bombs going off every second. This exerts tremendous amounts of upward pressure, against the enormous gravity of the star.
You do not actually need electron degeneracy pressure to resist the gravity of the star. It's like gilding the lily, or in physics term, mixing uranium with antimatter. It's useless.
Only later in the Sun's life, when the hydrogen runs out and the energetic Triple-Alpha process takes over, then the core of the Red Giant Sun collapses into a electron-degenerate white-dwarf seed with a density of 1000 tons/m3. Despite this, the core still has a lot of helium and some carbon in it. The Triple-Alpha process liberates a ridiculously high amount of energy, nearly as much as the CNO cycle that goes on in massive stars.
This means that while the core will be extremely dense (though nowhere near as dense as a neutron star), it would be so hot, that it would cause the surrounding material to swell up and become puffy, a Red-Giant. This white-dwarf seed is so hot that the Sun actually manages to fuse some carbon into oxygen, via alpha capture, where a carbon nucleus captures a helium nucleus and turns to oxygen. However this process is extremely slow as the Sun is simply not massive enough to fuse it on a large scale, it merely occurs as a trace reaction.
However let's move onto something that's dull and unimpressive compared to stars- Gas Giants
So then why would gas giant cores be electron degenerate, but not
become hot enough for nuclear fusion? Shouldn't they start fusing
before they can become electron degenerate, as stars do? Am I
misunderstanding this concept entirely, or is there more to it than
what I described?
Gas giants are called failed stars for a reason. They are simply not massive enough to fuse hydrogen to helium, you would need a core temperature of at least 3 million kelvins to initiate a proper fusion reaction.
However, do not underestimate gas giants. They are also called giants for a reason. Jupiter is literally so massive, that its core is crushed to a ball that is 20 times more massive than the Earth, yet has only 2 times the diameter. It is more denser than osmium (25,000 kg per cubic meter). This means that despite the fact that the core does not have the conditions adequate for fusion to occur, it is so dense that there has to be something that is exerting pressure from the outside. And it is electron degeneracy.
Put simply, Jupiter's core is a miniaturized version of a white dwarf, it is incredibly dense (though as dense as cotton candy to a white-dwarf) and incredibly hot (though it is next to freezing to a white dwarf). The core is crushed to such degrees that there has to be a upward push to prevent Jupiter's core from just turning into a black hole or something of that sort.
Basically, the Pauli Exclusion principle forbids two electron with the same spin to occupy the same "quantum state", which is fancy word for "place". So this principle is responsible for providing the degeneracy pressure that exerts a counterpressure against the massive gravity of Jupiter.
Stars are supported by thermal pressure as there is a lot of energy from fusing atoms together.
Gas giants are supported by electron degeneracy as they don't have anything else to hold them up.