That certainly could be done at some locations. But, where the published missed approach is 40:1-clear (2.5% clear) what would be the point?
And, unlike OEI contingency procedures, any alternate missed approach would have to have full TERPs or PANS-OPs lateral containment areas. And, unlike any OEI contingency procedure, the alternate missed approach would have to be accepted by the approving authority.
If the lowest minima are based on a 2.5% missed approach gradient then there is no need to do any more work. If however, there are minima provided for the 2.5% missed approach and lower minima provided for aircraft that can acheive a higher missed approach gradient, there may be something to be gained by revisiting the missed approach design with a few to making the route over more favourable terrain or even tailoring the procedure for example to the specific aircraft type that will be used (it might be at the lower end of the C/D speed range for example.
Yes indeed the alternative missed approach would have to meet all the PANS-OPS or TERPS criteria and not only that but when the publsihed missed approach involves a significant turn, one has to consider the posibility of an engine failure during or just after the turn perhaps when the aircraft has now turned towards the significant obstacles.
As far as obtaining authority approval, there would be no difference between this alternative OEI missed approach and the OEI departure. They are acheiving the exact same thing at different stages of the flight - allowing higher operating weights or in some cases operation in cases which would be impossible otherwise. Provided that the design is correct and the numbers work then safety is ensured.