Max thrust in an airplane with a constant speed prop will always be obtained at maximum RPM
Generally yes, and this is what you should expect, and operate to, though it is possible for the governor or fine pitch stops to be set a little too fine. I have experienced this a couple of times, and learned to set a slower RPM by experience, to get the greater thrust. For reference, when I did 39 precisely measured takeoffs in the C 182 amphibian for noise testing, I gathered the data to know. Most prominently, all other factors being exactly equal, the passing altitude at 2 km from brake release was 60 feet higher at 2700 RPM, than 2500 RPM. Yes, it was higher, but a 60 foot difference after 2 km, is bugger all. As MT had told me, run the prop more slowly, the thrust is the same. MT's STC on the 182 is at 2500 maximum, and I was aiming for approval at 2700, thinking there was bags more power. With the learning that day, I accepted EASA requiring 2500 RPM as a limit, for reasons of noise compliance. I had occasion to takeoff that plane from Split, Croatia, on a hot day. It was a sea level departure, but it was really hot, and I might have been one pound over gross. The performance was poor, and I had to fly down the coast for twenty minutes to gain altitude to turn inland over the coastal mountains. But safety was not a concern, as I had the sea to follow, without having to climb, if I could not climb.
The other factor, which is somewhat type specific, is for airplanes with a stabiliator, rather than a horizontal stabilizer, and elevator. Though I have experienced the problem in Cessna Cardinals, I have found the problem more pronounced in Piper Cherokees. What happens is a pilot desperate to get off the ground in hot and high conditions will lift off to early, with too much stabilator deflection. The plane is on the verge of being stalled, but will fly in ground effect. 'Problem is that the highly deflected stabiliator is creating a lot of drag itself' It's sort of stalled too, so really can't raise the nose any more (not that you'd want that anyway). but it seems to be enough drag to prevent the plane accelerating so as to climb out of ground effect. I was right seat in a friend's Piper Arrow when he did this on a very hot day. We were stuck in ground effect, and the plane was not accelerating any more. Though we had lots of runway (more than 3000 feet), it just kept going by. We were both so used to flying out of this runway, the alarm bells sounded too late. As he was drifting off the runway to the left, and landing back was obviously not going to be possible, I retracted the gear. Doing that was enough, that the reduced drag, allowed the plane to accelerate, and slowly climb out of ground effect. Alarmed by this experience, I borrowed a Cherokee during the winter, and took it out to the ice. With 20 miles of lake under me, and ten miles wide of "runway" I was able to experiment, and reproduce the effect, though it was neither hot, nor high. But I could stick it in ground effect, with landing back being the only option.
So, in Cherokees and Cardinals, soft field technique is okay, though understand your limits, so as to not get the nose too high!