In teaching a second year UK course in analysis some years ago I was surprised to find out how pleasant were the proofs of the some of the basic results using sequential methods, for example that a continuous function on a closed bounded subset of $\mathbb R^n$ to $\mathbb R$ is bounded. This inspired me to work out a paper on the notion of a $1$-point sequential compactification: add another point and let the sequences in $X$ with no convergent subsequence converge to the extra point! It got published too in the JLMS.
However for continuity of a function I do like to rely on the neighbourhood definition since a neighbourhood is a geometric object one can draw, whereas the $\epsilon - \delta$ are only measurements of the sizes of neighbourhoods.