My visual interpretation of $1+2+3+ \dots +n$
Not quite visual, but won't this be simpler?
Write:$$S=1+2+3+\dots +(n-1)+n$$ Reciprocate the order of terms: $$S=n+(n-1)+\dots +3+2+1$$ Add both: $$2S=\underbrace{(n+1)+(n+1)+\dots +(n+1)}_{n \text{ times}}$$ $$2S=n\cdot(n+1)$$
$$S=\frac{n\cdot(n+1)}2$$