He describes the style he did like as "Euler's style, so to speak". What was Euler's style in this context? i.e. as opposed to epsilons and deltas?

Euler was a master manipulator of formal expressions; don't think it bothered him very much whether, e.g., an infinite series converged or not. (Unfortunately don't remember a source for this offhand -- maybe a leftover impression from having read ET Bell's book long ago? I also don't know how reliable Bell is.)

Edit: added semi-source

Cauchy is the one who introduced rigor to calculus with the formalization of limits, including epsilons and deltas, a full century after Leibniz and half a century after Euler.