Just a few thoughts: Know that a teacher (and internet teachers) only know what they know and they don't know what they don't know, you know? We all think what we know and what we were taught is gospel truth.
First, the thumb isn't a finger and doesn't attach to the hand the way the fingers do. It is attached to the side of the radius and often when playing down, many teachers teach students to use the thumb's abductor which is its weakest and most sluggish muscle. The thumb's strongest muscle is its flexor but it curls the thumb under the palm and is designed for gripping. When the thumb is curled under the palm for many people it intersects with the index finger's long flexor tendon and can cause grinding damage. There are several disorders which can occur by curling the thumb under the palm such as distal intersection syndrome or Linburg-Comstock Anomaly. Also, curling the thumb under the palm creates a tension called a muscular co-contraction. Some teachers teach pianists to use the thumb through pronation or forearm rotation while others teach to play the thumb straight down using gravity or arm weight. Some advocate a tilt while others suggest a forward shift. I personally subscribe to using all of the above simultaneously to avoid any one muscle to be used twice in a row.
An example of teachers not knowing what they were talking about is Bach. He was a virtuoso whose technique came natural to him. He didn't know how or what he was doing so he taught his students what he felt and he felt his fingers caressing the keys so he taught his students the carrezando method. What he was actually doing was moving in/out/up/down and he felt like he was caressing the keys as his arm moved his hand across the keys, his arm was doing all the work and his fingers were not doing much and it felt like carrezando. Playing carrezando can facilitate flat finger playing which can lead to long flexor tendonitis and median nerve entrapment.
As far as the thumb being off the keys, all your fingers and digits are different lengths which SHOULD require the pianist to play with an in/out motion. Curling them to equalize them creates tension*. So when playing a five finger scale, the thumb plays and when the index plays the arm moves out because the index is longer, the middle finger moves the arm and hand further out. This will take the thumb off the keys then there is a shift forward to play the shorter ring and further in for the pinky. When the thumb is off the key, it should never drop below the keys and instead remain naturally and relaxed next to the index. When you allow the thumb to drop below the keys, it throws off any rotation and gravity of the arm and hand. Up is very important because you can't play down on the next note if your weight is hanging down. Your arm can only move in one direction at a time and if the thumb is pulling down the arm can't play up. It is like walking up stairs, your foot raises higher than the next step then comes straight down. If you don't lift your foot high enough you will trip up stairs. Dangling the thumb below the keys will throw the balance of everything else off. It also behooves the pianist to play on the outside edge of the keys because they are a fulcrum and thus, lightest on the edge and heavier the further in you play.
*With all five fingers together, wave bye bye. It should be effortless. Now curl and make a claw with your fingers and wave bye bye. You will probably feel strain. Likewise, abduct (spread out) all your fingers and wave bye. More strain. A lot of teachers will then tell their students they need to practice more, prescribe a silly exercise or tell them they need to build strength and endurance. I suggest they cease moving improperly and the strain will go away. Another fallacy is to tell the student to relax but, relax what? If you relax the muscles you are using to play, you can't play. Instead, relax the wrong muscles and employ the proper ones.