the engine out procedure is to fly runway heading
I think that the various previous posts have indicated that the original statement is not quite right -
(a) Type As and the like (generally, but not invariably) are straight out but they meet airport requirements, not operational performance
(b) Such charts provide some terrain information in a specific area but not information regarding the match of a specific Type to that terrain - that's a matter for the operator to figure out
as the engineers have analyzed the surrounding terrain for obstacle clear
Generally, this statement is not reasonable
Not sure how the original question was specific to takeoff
Most of us had presumed that to be the case. However, it really doesn't matter - the ground will kill you every bit as dead during the takeoff or the missed approach so there is merit in considering both. Logically, the takeoff is the more pertinent as every flight involves one whereas the miss involves much reduced frequency.
An EO for an approach requires a different set of performance criteria
Not quite. Approach and Landing Climb requirements are WAT matters and have naught to do, necessarily, with terrain clearance. Any specified missed approach gradient requirements generally are regulatory and related to procedures design rather than EO escape.
More particularly, none generally has any interest in the clean up phase.
Rationally, the only practical way to address the missed approach terrain clearance is by adopting an implementation of the takeoff data. While this presents some interesting bits, it is generally not a major difficulty, especially with the more reliable tracking information the black boxes can now provide.