8che,
unfortunately our aircraft are not catIIIb and as such cannot perform CAT III one engine out.
Yes you can have multiple scenario but the take off alternate is chosen to be within one hour one engine LRC. If you have an hyd problem, electrical problem you can still fly on 2 engine and as such will cover more ground that on one engine.
An enroute airport is still called alternate as per Air Ops and as such during dispatch you would still need to cover for the dispatch criteria either 1,67 or 1,92.
While I agree the take off alternate is mostly due to weather, landing on engine would be worst scenario as you need to land at the nearest suitable airport. You would not spend time lowering the fuel before landing. If you had flight control or gear problem, you would fly more longer as to reduce your landing weight by burning fuel (and such cover more ground to a suitable alternate)
But whatever scenario you take, the question was do we need to consider dispatch criteria or not. Looking at Air Ops and the reply from B737900Er, it is an alternate and as such you need to use those rules.