Why is the definition of "limit" difficult to understand at first?

As far as I see it, there are two massive obstacles that a classroom must overcome when trying to come to grips with formal limits.

A big problem with understanding the limit definition, as you have pointed out, is the problem of quantifiers. There are so many, and the order in which they are employed is so important, that it can be difficult to keep track of. (And just when they think they understand, now they have to deal with uniform continuity, which to most will look exactly the same as continuity!)

To make matters worse, it's hard to describe a limit in a language that is precise enough to be useful to solve problems, but gentle enough to be swallowed easily. This is hardly surprising; literally thousands of years of mathematics was done before anyone really found the need to understand them completely.

A big problem with using the limit definition, is that it feels circular. This is often the first time students will have to consciously work ''forward'' to first convince themselves what the limit is before they can prove it.

For example, when you are solving $3x+4=5$ there is a mechanical list of procedures that one can go through to get to the answer. Even when doing something more technical, like verifying the trigonometric identity $\tan^2x + 1 = \sec^2x$, you can still start from what you're given and come up with the answer. Thoughtful, informed students will know this logic needs to be run in reverse to constitute a proof, but there is at least a foothold for one to grab onto; that original equation.

But when you are trying to find the limit of $x^2$ as $x\to 4$, if you don't have the intuition that the answer should be $16$, your work becomes very difficult. You can apply the limit definition to try to find a $\delta$, but the catch-22 is that you need to ``know'' what the limit is before you can even interpret which points are in the $\varepsilon$-neighborhood..

I believe the problem with introducing limits is that the quantifiers might be confusing for students looking at it for the first time. Instead of a symbolic definition, it is important to explain it pictorially, to begin with, and then move on to the "formal" version. I believe "seeing" something pictorially enables the students to understand the key ideas succinctly and conveys to us a depth of understanding that a symbolic description cannot easily match. John Morgan Allman, in his book "Evolving brains", claims that evolution has worked $1000$ times longer on our perceptual abilities than our symbolic-reasoning abilities. A simple example to start with would be say to consider a sequence $$x_n = 1-\dfrac1{2^n}, \text{ where }n \in \mathbb{N}.$$ You can ask the students to mark these points on the "real" line

and hopefully they draw something like the above. They should now believe that the sequence is less than $1$ and can be made arbitrarily close to $1$. Once they have this picture in mind, you can have a formal definition of the limit of the above sequence.

You can then ask the students to consider different real sequences and ask them to mark the real sequences on the "real" line. For instance, a real sequence that converge to $1$ but oscillates around $1$. Once they have played around with marking these real sequences on the real line, you can then introduce the symbolic definition, with the different quantifiers, to define the limit of a general sequence.