Surprisingly elementary and direct proofs

Goedel's original completeness/compactness proofs in logic are very hard and technical. Modern versions of the proofs are considerably simpler and do not use any sophisticated new theory.

Most proofs of the fundamental theorem of algebra use some topological/homotopical deep(ish) results or some deep(ish) results of complex analysis. In fact, the theorem can be proved using only the definition of complex numbers and absolute value, using very elementary properties of complex numbers, and the entire proof is about half a page long (such a proof is actually a minimum modulus argument but applied to a polynomial so that the computation can be done directly without appeal to the general minimum modulus principle). The proof is extremely elementary.

The proof that Euclide's fifth postulate is independent of the other axioms of Euclidean geometry by exhibiting models (i.e., the sphere and the hyperbolic plane) where the postulate fails (in different ways) is completely elementary. Certainly all those who tried to prove/disprove the fifth postulate did so while walking on a counterexample. However, the numerous attacks on the problem prior to its settlement in the 20th century were quite sophisticated and laborious. The barrier was conceptual - the understand of the importance of models - not technical.

Brouwer's topological proof of his fixed point theorem used (I believe) homology and/or the fundamental group. A perfectly elementary proof using Sperner's Lemma was later discovered (I'm not entirely sure about the chronological order here, so I may be mistaken).

Initial proofs that the higher homotopy groups are all abelian were greatly simplified by the Eckman-Hilton argument (which is completely elementary).

The Mac Lane coherence theorem in category theory is rather technical and is enormously simplified by considering the Yoneda embedding in the 2-categorical setting. All of the machinery was already present, but it required some re-assembly to figure out a rather elementary proof.

Gauss's initial proof of quadratic reciprocity in Disquisitiones was exceptionally long spanning I believe 30+ pages, and in my opinion it is incredibly nasty. Gauss used induction and then reduced to a something like eight cases. While not necessarily relying on big theorems this proof seems overly complicated especially when compared to some of the proofs that followed.

For example, Eisenstein's proof is quite short and has a very nice geometric component that I think is very nice, and it is does not require a great deal more machinery than Gauss's original.

There are in fact may other nice proofs of quadratic reciprocity. Here's a nice list of some of them

Chebyshev had first proven the Bertrand's postulate which states that for any integer $n > 3$, there always exists at least one prime number $p$ with $n < p < 2n − 2$. A weaker but more elegant formulation is: for every $n > 1$ there is always at least one prime $p$ such that $n < p < 2n$.

Later Paul Erdős had given an elementary yet elegant proof of this. Surprisingly he was only 20 years old when he had given this proof.

Too long for a comment.

I think Bill Johnson's answer (Lomonosov 1973) on MO applies to your question as well. Some superfluous theory that was used for the first proof of a weaker result (Bernstein-Robinson 1966) was nonstandard analysis. It was Halmos who immediately removed the nonstandard analysis from the argument.

These things were certainly not easy to figure out because von Neumann himself only proved the existence of nontrivial invariant subspaces for compact operators on Hilbert space ($\dim \geq 2$). Lomonosov entails hyperinvariant ones on general Banach spaces.

See this wikipedia page on the invariant subspace problem for statements and a chronology of related results. Maybe that's a good opportunity to recall that it is still not known whether every bounded operator on an infinite-dimensional separable Hilbert space admits a nontrivial invariant subspace.