I think the generational tactical shift underway in the Premier League is going to make the combination of speed and ability on the ball more decisive than it has been for a while. We've gone from a handful of clubs employing high-energy pressing with success, to a situation where any club that can manage it at all presses like crazy, even the new promotions from the Championship.
The change is already making certain kinds of players, for example more static or crafty forwards, or centre halves who are useless on the ball, far less desirable to top clubs.
This piece from the Athletic (paywall circumvented) was very interesting:
https://archive.ph/zF5l0
To be successful with old school counter-attacking tactics, the qualities you look for in players are precise and visionary distribution from the keeper or the deep-lying playmaker, and then strategic runs, offside trap awareness and pace to burn from the forwards. The forwards running onto a release pass rely on their composure, heads-up awareness, and finishing ability to score.
But in the case of one of these "direct attacks", though the accuracy and tempo of the release pass is still vital, the forwards are operating with less time and space having just jumped a well set high defensive press, and needing to work collectively to beat defenders, drag defenders way out of position, or use the yards between recovering defenders with speed and precision to score.
Given foot speed for defenders is also at a premium, none of these aspects is going to get any easier for forwards. They're going to need to be fast, smart and strong in all technical aspects. A player like Martinelli is the model, whereas things might get a bit tougher for forwards with amazing weapons but also certain technical limitations, like your Walcott or Agbonlahor who used to be able to prosper in a counter-attacking Premier League side.
In today's league, every top club is going to proactively seek out players that can gazump a high press in this way without first turning over the ball and then hoping for a productive counter-attack.