For each $N > 1$, let $A_N$ denote the set of all positive integers that are not multiples of $N$, and let $a_{N,1} < a_{N,2} < a_{N,3} < \dots$ be the elements of $A_N$ in increasing order. The set $A_N$ consists of blocks of $N-1$ consecutive integers followed by a skipped integer. If $N \equiv 1 \pmod 3$ the length of each of these blocks is a multiple of 3 and so $a_{N,i}$ can only be the last element of a block if $i$ is divisible by 3. In other words:
$$\text{If } N \equiv 1 \pmod 3 \text{ and } i \neq 0 \pmod 3, \text{ then } a_{N,i+1} = a_{N,i} + 1. \quad (1)$$
For each $N > 1$ that satisfies $N \equiv 1 \pmod 3$, we will show that $A_N$ is a special set. Since $2023 \equiv 1 \pmod 3$, the desired result follows.
We now fix $N > 1$ satisfying $N \equiv 1 \pmod 3$. For simplicity, from now on we suppress the $N$ index in $a_{N,i}$. We will use induction to show that for each $n \ge 3$, the numbers $a_1, a_2, a_3, \dots, a_{3n-3}$ can be used to form a special $n$-triangle.
The base case consists in showing that there are special 3- and 4-triangles.
$$\begin{array}{ccccccc} & a_1 & & a_1 & & \\ & a_6 & a_5 & & a_6 & a_9 & \\ a_2 & a_4 & a_3 & a_7 & & a_5 & \\ & & & a_3 & a_4 & a_8 & a_2 \end{array}$$

When $N \ge 10$, $a_i = i$ for $1 \le i \le 9$, and it is easily checked that these triangles are special. For $N = 4$ the above turns into

and when $N = 7$ we get

and these are indeed special triangles.
For the inductive step, we show how to obtain a special $n$-triangle for $A_N$ from a special $(n-2)$-triangle for $A_N$. Any special $(n-2)$-triangle contains the numbers $a_1, a_2, \dots, a_{3n-9}$. We group the next six numbers of $A_N$ into two sets, $S_1 = \{a_{3n-8}, a_{3n-7}, a_{3n-6}\}$ and $S_2 = \{a_{3n-5}, a_{3n-4}, a_{3n-3}\}$. By virtue of statement (1), both sets consist of three consecutive integers. Hence,
$$a_{3n-3} + a_{3n-8} = a_{3n-4} + a_{3n-7} = a_{3n-5} + a_{3n-6}.$$
It is now clear that we obtain a special $n$-triangle when we add $a_{3n-3}$ and $a_{3n-8}$ to one side, $a_{3n-4}$ and $a_{3n-7}$ to another, and $a_{3n-5}$ and $a_{3n-6}$ to the final side in any given special $(n-2)$-triangle.