Most pilots today fly by numbers. It makes sense to train that way because, as one moves up to heavier and higher performance aircraft, procedures and calculated numerical values become more essential to safe flight. Airspeed control is stressed by the numbers. Coordinated flight is taught by instrument reference. ("Keep the ball centered with your feet.") And while some attention is paid to getting an aural/tactile feel ("Learn the sound of a good glide"), seat-of-the-pants flying is mostly a historical reference.
But let's say that you don't matriculate to a Lear or a Boeing, or like me, you move back to become a journeyman VFR lightplane pilot. Do you hone your proficiency by tightening up the numbers? Nope. You move away from the digital and bring in the analog. You leave your eyeballs outside, where they belong in visual flight, and you turn over the management of the cockpit to your other senses.
Find an experienced ag pilot, and ask him the V speeds of his trusty flying companion. Chances are that he won't know them because, while his type of flying is arguably the most precise of all, it's not quantified numerically, but is rather a balance of sights, sounds, pressures, and even smells. To achieve the necessary precision, he must use his brain naturally, like an athlete. Numbers and symbols are reserved for the invoice.
I've never seen an agplane with a "ball" indicator. While coordinated flight is a life/death necessity with a heavy load, so is obstacle clearance. Precise VFR pilots have perceptive rumps. Backside and seat are analogous to ball and cage. A little practice develops this sense.
Try watching the ball for a while during cruise in light turbulence. Note that displacements of the ball are accompanied by corresponding pressures felt through the seat. Applying rudder pressure to keep the ball centered will keep your gluteus maximus centered, and vice versa. Learning a relaxed posture is often necessary. Rudder should be applied by moving the ankles with calf muscles. "Booting" the rudder or just plain "pucker factor" tightens the gluteus maximus muscles, altering sensation.
Hand and foot pressures are good indicators also. My favorite way to quickly develop the feel of an aircraft is to do Dutch rolls at various speeds and power settings, keeping the ball/buns centered. In a tractor single, empennage effectiveness varies with changes in speed and/or power, while ailerons vary only with speed. Realizing this and paying attention to the pressures needed in various regimes develops an intuitive smoothness. It's very satisfying to be able to roll into and out of maneuvers without being jostled.
The ear can be a very precise airspeed indicator, reading not in numbers, but in margin above the stall. Do a stall series in a light single, and note the sound changes that occur in increments above the buffet. My 172 wheezes around the fresh-air vents a couple of knots before the party-favor-style stall warning begins to hum softly, about 2 knots before it enters full song, some 3 knots before the stall. Knowing these sounds lets me fly repeatable, steep, short approaches with my eyeballs on the obstacles.
Developing and combining visual/aural/tactile senses beyond what we got from the training syllabus lets us fly more by instinct than intellect, which I believe to be the best way to go VFR.
At work, we fly Schweizer 2-32 sailplanes carrying passengers (mostly couples) on sightseeing rides. The 2-32 achieves its good performance with a 57-foot laminar-flow wing having somewhat twitchy stall/spin characteristics. Circling and skimming ridge tops in heavy traffic leaves little time for gauge watching, and we often can't afford to carry extra speed that would end the ride prematurely. We keep in mind that a wing can be stalled at any speed but also that it can be flown at very low speed, provided that it is not loaded to the point of airflow separation. In other words, it can't stall if you're not pulling on the stick (except in heavy turbulence). So we pull Gs in relation to the sound of air on the airframe.
We're motivated to make greaser landings ("Gratuities are accepted") and also to hit a spot that allows us to roll to our launch area, avoiding laborious "taxiing." I prefer to fly a 180-degree curved approach from a close-in downwind to threshold. My side-by-side-mounted eyes give good lateral depth perception but see more like one eye in the vertical plane. The final turn gives an ever-changing perspective that helps me use the spoilers to hit my spot each time and to kiss the rigidly mounted main wheel onto the chip-seal more often than not.
As I write this, a strong, gusty wind streaks the treelined pond outside my window. Something has died in the watershed; three ravens and a turkey vulture are making their moves. Quickly, the smaller ravens with their superior STOL capability land and hop toward the carcass. The big buzzard is operating in tight quarters. He circles the half-acre pond concentrically, making huge, perfect drift corrections. He instinctively modulates his wing as it passes through myriad treetop vortices, all the while rapidly swiveling his shiny, red, bald head as he scans traffic, obstacles, touchdown zone, and Beo, our pet wolf. At last, he drops in steeply and flares to a three-step rollout. What an approach — this guy has mastered the medium. That's the way I want to fly.