Some considerations ..
(a) as part of the certification process, an aircraft will have weight limits which address a set of required WAT gradient limits. These vary with the number of engines and look both at a reasonably to be expected gross capability and a terrain clearance lesser climb gradient to use when running obstacle clearance sums for the OEI escape.
The important thing about WAT limits is that they represent a line in the sand set of performance criteria so that the aircraft is likely to be able to keep going up when that is required. If your weight is less than the relevant WAT limit weight, then your performance, pretty obviously, will be better than that which might be achieved at the WAT limited weight.
(b) for the more common two-engine aircraft, for the second segment as an example, the minimum gross gradient required (which will apply if your takeoff were at the WAT-limiting TOW) is 2.4%. With the two-engine margin factor of 0.8% applied, the net gradient required is 1.6%. Both of these can be converted to an expected ROC using the usual trigonometric sums while keeping an eye on whether the air is non-ISA for any instrument considerations so that we can reasonably match the numbers to the rocky bits.
(c) Multiplying by 60 is just a rule of thumb Generally, if you are running sums involving angles, the 60 factor comes from radian angular measure conversions where the appropriate number is around 57 (if my memory isn't failing too much .. =180/pi). That's pretty close to 60 so we use 60 and accept any small errors associated with the calculation.
(d) what distance from the runway, according to certification standards, does the gradient requirement change from approach climb (2.1% for a twin jet with OEI in approach configuration) to landing climb (3.2% AEO in landing configuration)? As another poster observed, the figures you quote are certification WAT gradient requirements and, as such, provide a line in the sand for maximum weights on the day. They have nothing specifically to do with managing the approach and landing but will be incorporated in configurations for the approach and landing and, more particularly, the missed approach.
(e) we need to keep in mind the consideration that the WAT limits provide for the maximum allowable weight in whatever situation you are looking at. Calculating numbers based on WAT limits will be relevant if you are at the relevant limiting weight. If you are lighter, then you will perform better and the calculated numbers will be conservative to whatever extent. More important to remember that the AFM performance data incorporates all the various WAT limits .. the important thing for the pilot is to fly the aircraft as the AFM expects so that there is a reasonable expectation of achieving or exceeding the climb capabilities incorporated in the AFM data.