They (FAA originally) deviced these manoeuvres and set the requirements
Not quite right.
The original story for the current performance philosophy/requirements is the ICAO Final Report of the Standing Committee on Performance dating back to 1953. One of our posters, a long time colleague and very kindly, passed his personal copy onto me some time ago .. now a valued reference library document on the bookshelf to my side.
Various States then publish their implementation (evolving over the years) of these requirements. Hence we see FAR 23/25 etc.
It has nothing to do with obstacle clearance. They demonstrate that the aircraft has a certain climb performance, nothing more, nothing less!
Not quite right.
(a) you are talking about WAT limits .. which are included in the AFM and represent the Design Standards bottom climb line. Rarely are these the critical path for a takeoff weight calculation in the field. Usually, either runway length or obstacle clearance is more critical.
(b) you are overlooking segment obstacle clearance .. which also is included in the AFM and must be met at planning to establish RTOW compliance.
A complete different authority (ICAO), set a limit for minimum obstacle clearance
Not quite right.
PANS/OPS relates to routine AEO operations, letdowns, etc. Nothing to do with takeoff calculations which are AFM based and OEI.
In short, so even with gear-down and accelerating, your modern aircraft should have a gradient well above 3.3%
Now that depends entirely on the gross weight and how the aircraft is being operated .. a very long bow to draw, methinks.
In VMC, terrain in sight, you can go on according to what you see.
Danger, Will Robinson ...
OEI, the gradients can be so shallow that they defy any perceptual attempt to wing it on the day.
Either you have done the sums and fly to the procedure .. or you are in the lap of the Gods .. and the Gods, generally, don't care a fig, ergo, a high probability of a subsequent CFIT.
Some States have permitted this sort of thing for smaller heavy aircraft .. eg, Australia for under 50,000lb permitted reduced splays in VMC (to allow the F27 some flexibility in the early days). Interestingly, this was metricated to 20T (if the memory serves me correctly) .. which thwarted the original rationale. I protested formally (long ago) and gave up in the face of the usual opposition. I notice that it now has sneaked back up to 22.7T ...
Main concern is that it just is NOT feasible to wing OEI obstacle clearance visually ... requires visual/perceptual discrimination beyond our capability.