Every ETOPS operation I've been involved in has at its heart, struggled to explain in clear terms this little misconception as previously described.
From a simple perspective ETOPS is a planning exercise with requirements for improved safety margins and weather minima etc. Once dispatch ( that's another can of worms!) has been achieved then all that needs considering is the standard weather minima...like you normally would on a non ETOPS flight with respect to alternates. The only ETOPS specific consideration would be to ensure that prior to the ETOPS Entry Point the weather at your ETOPS alternates is above the normal minima, if not other alternates are required...but at normal not enhanced minima.
That's how it's always been done where I've worked.