Shorter intervals on a High Mileage or cocktail blend with anything; yes even full synthetics, is going to potentially get more out.
Remember, this isn't a matter of "for the oil's sake", but rather for the engine's sake. While it is debatable I certainly wouldn't count on conventional to clean an engine with that many deposits as the filter is showing. At least not as the most reliable method.
Therefore, a HM oil; MaxLife full synthetic from wal-mart!
+ a Q of MMO or the 5w-40 Rotella T6 with MMO replacing a Q instead is a good COA if cleaning the engine up over the next year is the goal.
I would not be extending the interval so even if somewhat of a waste it's not going to hurt it to use synthetic for the cleaning interval; whether it be a HM synthetic or top-tier just don't use conventional at 5k thinking it'd clean it fast enough only to realize it can't hold the contaminants in suspension as well as a HM or full-synthetic.
The engine oil filter is not designed to clean up the sort of crud dirty engines get filled with. At least not beyond a point, which again is why shorter-OCIs are generally the first step in cleaning an engine.
If the engine was clean, sure it'd be pointless to change a full synthetic or even conventional out early. But the issue with cleaning the engine is not 'value' in oil price, but rather accomplishing the most cleaning as possible which is a bit of an unknown still.
The bare minimum is conventional at 3k intervals, but really with an unknown of 'how dirty' that is a 'get by' approach without spending more for a regimen that at least gives a better chance for more aggressive cleaning, IMO.