Here is one of those writers who claim that lift is smaller in a climb than in straight flight.
With good reason.
The sum of all vertical forces will be zero in a stationary climb or the aircraft would accelerate. It is a fallacy to assume that lift is higher at higher speed – lift is just as large as is needed to equalize all other forces. Now it is important to note that the aerodynamic coordinate system (in which lift is defined as the aerodynamic force parallel to the Z axis) is tilted, pointing slightly upwards with its X axis parallel to the flight path. Drag is slightly pointing down and lift is slightly pointing back when seen from an earth-fixed observer.
To equalise those forces, we have only thrust (at least in the absence of wind – or this would be the special case of a glider climbing in a thermal). This thrust must now compensate both the drag force and the backward component of lift. It is, therefore, higher by that backward component of lift compared to straight and level flight. And it is also pointing slightly up due to the climb attitude of the aircraft. Therefore, this upward-pointing thrust component (which is larger than what is needed to compensate the downward pointing drag component) works against gravity and helps the lift in compensating weight. Lift can, therefore, be a bit smaller than in straight and level flight.
Forces and their angles acting on a climbing aircraft in side view. The light, translucent arrows are copies of the solid arrows shifted into a vector chain to show that they indeed equalize to zero net force.
The extreme case is a vertical climb: Now all the force counteracting weight is supplied by thrust and lift is zero, or the aircraft would accelerate sideways. Climb is simply a linear combination of level flight and a bit of vertical climb.
Now you could argue that the B-52 climbs with its nose pointing down, so thrust must also point down a bit. Maybe, but in level flight that thrust would point even more down and more lift would be required to compensate for that downward thrust component. No matter how you look at it, as soon as the flight path angle pitches up, so does the thrust, thereby reducing the required lift.
Now you argue that climbing at a faster speed is better. But that would create more drag, leaving less excess power for climbing. Also, when flying faster, the aircraft must reduce its angle of attack in order to produce the right amount of lift. Climbing is not about lift maximisation, but about excess energy maximisation.