Apologies. I owe you an answer to your question from a while back.
First off, I want to state that I don't know what I'm talking about! I have ideas and conjecture but nothing that's definitive. Please bear that in mind as you read on.
Okay, engine oils to clean up the mess caused by other engine oils isn't that well researched a subject. In fact I'd go so far as to say it's not been properly looked at at all, ever. Think about it. Why would the oil companies look at such problems when to do so, they would have to acknowledge that might be something potentially wrong with their existing juice in the first place? This is anathema to them!
The oil companies do make some impressive claims for cleanliness and clean-up for some of their top-tier stuff but IMO you really need to take these claims with a big pinch of salt. As often as not, they refer to sludge rather than deposits and the demo tests they use are also somewhat 'artificial' in that the go though a rapid lay-down phase followed by an immediate and rapid clean-up phase. This is not the same as removing baked-on gunk from a piston that may have built up over several years.
To make this containable, I'm going to assume you've read my old posts. You will therefore already know I'm rather critical of certain aspects of modern engine oil design and why. You will have seen that I bang on relentlessly about the dangers of high Noack (especially in the US). Noack is like smoking cigarettes, you can smoke and live till a hundred or you can die of lung cancer at 35. Likewise you can rack up 300k miles on a high Noack oil or you can find the engine consuming vast amount of oil at 50k miles usually due to late onset oil ring stick. You will know about my concerns about base oil solvency. You will have heard me go on about how fuel dilution, more prevalent today because of GDI, can interact with Noack to worsen 'over-the-top' deposits. You will have read about my criticism of the API and the glaring gap in the testing protocols regarding deposit build-up over more than one OCI and why the OEMs are less than interested in excessive oil consumption if it only kicks off outside the warranty period.
So what's to do? First off, to those members who can drop an engine, strip it, remove all the pistons, clean then and put all the bits back together in a weekend, I salute you. Likewise those members that pull spark plugs and do piston soaks with MMO and the like, I salute you too. But I for one can't do that stuff. I never could. I haven't wielded a spanner in anger in 35 years and I don't do messy. HOWEVER...if I did have a car where deposits were playing havoc with the engine, I think that even I could source a 'special oil', drain the old old, replace it, drive it clean for say 1000 miles and then dump & drain. This is what I think I hear on BITOG; the relentless scream of demand unfulfilled!
What would such an oil be? First off it should be heavy and aromatic. The closest you might find would be a low tier Group I 20W50. You don't want Group II. Ideally it would be VII-free, something like a 20W40, but these are rare.
If I could go beyond the normal, I might think of an oil that contained say 30% of 'Solvent Extract'. When you make Group I base oils, you take crude Vacuum Gas Oil (VGO) and you mix it intimately with a solvent like Furfural. The mix separates into two layers. One that is 'almost' high VI paraffinic Group I base oil, the other a mix of Furfural and low VI, unstable heavy aromatics. The former you send to Dewaxing, the latter, you separate by distillation with the Furfural being recycled and the so-called 'Solvent Extract' going somewhere not lubey (fuel oil blending? FCCU feed? don't know to be honest). Anyway, whilst Extract is conventionally seen as very bad base oil, for the purposes of piston cleaning it might be just what you're after. It's also dirt cheap and The Sovs still make thousands of tonnes of this stuff so it's available. Just bear in mind you are looking for a cheap deposit cleaner that can be functionally used as an oil for a very short period.
That's probably enough said to get the ball rolling...