If stability will be the aim for him for the first couple of seasons, why not for others? I don't see why he'd have different aims to, as you said, Pellegrini, when backed with the same resources. Pellegrini may have achieved higher league positions at City, but he'd inherited a side that recently won the league.
We achieved top 6 "stability" under Moyes consistently on a shoe-string budget. I would regard it as a negative thing if we were to do no better on a large one.
Because his pedigree isn't one of winning major-league trophies (bar one domestic cup with Valencia). His recent pedigree is stabilising Southampton into a decent league position. He didn't win them a cup but they might yet just about qualify for Europe.
De Boer has no major-league experience, so we can't realistically think stability would be the aim as he's an unknown quantity.
Pellegrini has won the Prem and will be expected to get us top 4: we'd be pushing for quick progress under him, not working towards stability (which would then come naturally anyway through Champs League qualification). Sounds great, but then there's a risk involved as some of the City performances have shown.
Koeman is the
stability option to return to where
Moyes had us. Considering how much ground we've lost under Martinez, this is the sensible option.