Burnes is averaging a strikeout per inning. Once upon a time, that would have been considered elite. In today's game, that's the status quo so I'm not surprised there's some knit picking about it. But it's good enough for me and there's nothing wrong with pitching to contact when you have a very good defense behind you. Trying to strike out everyone just piles up the pitch count.
Strikeouts in and of themselves do not lead to more pitches being thrown. Walks, hits, hit batters, and errors are more likely to lead to increased pitch counts than strikeouts. Having said that, I do agree with the first part of your last sentence - “trying to strike out.” Erik Bedard used to frustrate the heck out of everyone because he could easily get to 0-2 on a lot of hitters without really trying too hard. But then you could count on the next pitch, and sometimes the next 2 or even 3 pitches being low and away and out of the zone, which batters quickly figured out they’d be better off taking. After a game in Seattle Ray Miller went off on him in the post game presser for throwing way too many pitches in an effort to get strikeouts. He went on quite a tear shortly after that where he mostly stopped doing that. In that Seattle game, it actually seemed like a lot of their hitters were taking 0-2 pitches because they knew what he was going to do. He went into that game with almost no-hit stuff, but ended up getting pulled in like the 4th inning because of the pitch count. I’ll see if I can find the box score to verify my memory.