He may well have had, but I doubt it was quite the same flavor as in the Stan and Jack years. I've always thought of what Spidey has as more like 'flippancy' than 'cockiness'; all the various deaths he's had on his conscience through not being fast enough to save someone (even back in 1970, he'd racked up a few) mean he's all too aware of his weaknesses. The first few years when he was a teenager, he perhaps had a certain flavor of cockiness to him, but it quickly became more of a tactic to distract his enemies during battle than a genuine vanity. 'As long as I'm around, you'll never be the best' just seems out of character; he'd be more likely to make a joke of some kind.