I've quoted: FAA, TC, CAAS and several company OPS manuals. I also included the references.
Henry was kind enough to quote EU-OPS.
The general point that you seem to be missing is this:
All flights must operate in a safe manner, and therefore have a place to go in the event that things don't go as expected (Common sense amongst other things).
Some regulations include this requirement. Some don't. Many - no most -western Operators include a similar requirement.
Non-etops flights (When required) have the same weather requirement for enroute alternates (weather.....1 hour before ETA to 1 hours after...etc), along with destination alternates, take-off alternates, etc. The difference is 60 mins, versus 120,180, etc, etc for ETOPS.
Also, there is the inclusion of the term "Suitable" with ETOPS, which is not included in Non-ETOPS. Non-ETOPS simple says, "Adequate Airport...with "weather above landing minima..."
Now, if your ME airline believes that flying "10000 miles" with no "Adaquate Enroute Airport" with "weather above landing minima" is acceptable, then rest assured I'll be avoiding that airline!
Big reminder! We are talking about "Planning" not about "In-flight". And, as BOAC stated
When it is, those weather restrictions apply