He mentions that Vx is close to stall speed - ever heard of Take Off Safety Speed? The manufacturer states this is the minimum speed until 50' to ensure controllability of the aircraft in the event of an EFATO.
Most manufacturers POH for light aircraft refer to lift-off speed and 50 foot barrier speed ( which is usually close to Vx). There is no mention of Vtoss.
The Bonanza POH states "set desired climb speed once clear of obstacles", assuming the obstacles were deemed clear on lift off there is no need to fly Vx or Vy.
Piper breaks up take-off into take-off and climb; for take-off they recommend accelerating to Vx to clear obstacles then Vy or just accelerate Vy if no obstacle, something to note is they do not say to accelerate and maintain these speeds, they are implying a minimum speed. En-route they state the Vx and Vy speed but usually have a recommended higher speed. They do not state where take-off ends and en-route climb begins, one can assume that is at a safe altitude once obstacles/terrain is no longer a threat.
From the PA-32R-300 POH (1987 copy);
"Allow the airplane to accelarate to 52 to 62 KIAS depending on weight of the aircraft and ease back on the control wheel to rotate to climb attitude"; end of normal take-off procedure.
Climb procedure next; "The best rate of climb at gross weight will be obtained at 92KIAS. The best angle of climb may be obtained at 87 KIAS. At lighter than gross weight these speeds are reduced somewhat**.
For climbing en-route, a speed of 104KIAS is recommended"
So based on that if obstacles are not an issue the manufacturer is basically saying lift of and accelerate to 104 KIAS for a normal take-off, Vx or Vy only if required.