The problem, always, will be that the escape flight path can be quite variable from one runway to another.

While there is a climb gradient penalty with turns, and this increases with bank angle (which is why the default maximum is 15 degrees to keep things under some sort of control) the turns are required, in general, to avoid the rocky bits out there. Keeping in mind that the rocky bits might be on the inside, the outside, or both sides of the turn, and that the required bank angle is tied to the turn radius and speed, both bank and speed are critical in tiger country departures for terrain clearance .. not to mention getting the start turn point correctly flown. Generally, the designer will include some flexibility to accommodate the realities of flying the departure and these parameters should be declared for pilot information. However, at the end of the day, an OEI escape in tiger country is an extremely critical and demanding flight manoeuvre.

If the automatics are up to flying to the required accuracy, there is no philosophical reason why an escape can't be programmed into the database. Indeed, if that is feasible, it provides for a better solution than getting the human pilot to fly the escape. One of the problems will relate to whether the particular system will permit pilot programming to the required accuracy or not.

what would be the point then of airbus providing the EOSID feature? it even automatically makes a stored EOSID to that runway (solid yellow line) the temporary flight plan if an EO condition is detected and if it happens before the diversion point. (DSC-22-20-60-40-A)
of course, you need to monitor your guidance and take care of any other ristrictions (i.e. bank angle, etc) which are mentioned in your EO plan. but that's no different to selected guidance.

On a slight tangent, my airline has disabled the EOSID function. We have to load them manually in the SEC FPLN. Can't figure that one out.

In the case of Kathmandu, the company had specific engine EOSIDs, yet we had to disconnect the autopilot in order to be able to bank 30 degrees and remain in the containment area.

Please do not confuse FMGC stored EOSIDs, which are appropriately validated by the Operator as part of the database, with the crew manually entered engine out procedure waypoints. This thread, as far as I understood, deals with the latter and the restriction that some Operators impose of selected guidance due to a CFIT risk assessment should the waypoints be inserted inaccurately and/or the "path" be flown inaccurately by the aircraft in NAV.

EXCATLY!!!!
So where is the reference for this in the manuals?
Or is it as your state a, "restriction that some Operators impose"?
The reason I ask for an Airbus Reference is that there is nothing in our Op's Manual prohibiting this. And ALL instructors claim it is an Airbus Limitation.