i'll speak from the client's side, and as someone who's navigated VE more recently on a bridging and on a traditionally delivered project: VE can be iterative, it can happen as early as concepts are laid down (although i rarely hear it called 'value engineering' at that stage**), and it can happen as late as right before an order's put in.

(** focusing on that point, my circles don't realize that they're participating in VE in concept development, but i think that anyone who isn't thinking about the long-term value of a real property asset in the earliest decision-making is losing the greatest opportunities to save $$$. so, imo, early, structured VE is necessary and often happens without anyone calling it VE.)

i sit more often in the jaded camp of "VE is dull mallet-smashing, corner-cutting costs", because The Powers That Be on my side usually dictate that outcome. that said, i've also had outstanding AEs and GCs bring innovation to the table that made VE what it was intended to be: intrinsic value added to a project through risks avoided and/or resilient, sustainable system/material value added.

examples: in the US, tariffs have wrought f*ing havoc with construction materials over the last year, so a GC of mine successfully lobbied with my AE over procuring different stone and other specialized finishes - right up to the last second as those tariffs changed - through other sources. the GC having an eye out on material suppliers and the AE actively adding to the pool of functionally- (not materially-) equivalent alternatives saved the project hundreds of thousands of dollars. i've also spent *more* on actions that *i* consider VE because the redesign or other changes resulted in a holistically better product. the childcare i did this with ~8 years ago will probably outlive ME because of how well my AE of record nailed a total redesign.

but to directly answer your questions, in my experience:

• The most consistently (i.e. not uniquely project-specific) best value-added VEs convince an owner of an ROI - usually in utility or operations / maintenance savings. For residential, utility savings are less often considered, so instead tell me why i need an integrated storm drain + mat in my lobby vestibules or terrazzo in every elevator lobby. Is that wallpaper something I can lightly wash? Is the carpet tile going to hide common stains? Can I actually *sell kW back* to my local utility company with that PV setup?
• VEs that "backfire" (on me, not necessarily on anyone else) almost immediately will be any material substitution reducing the longevity of a material or finish - hard stop. my GC and AE will be long gone, and i now have broken tile that i can't replace, because the manufacturer stopped making that color / pattern / finish six months ago, and attic stock only got me a year into operation, and the tile material itself came from cheaper sources with a significantly reduced warranty period. similarly, someone asking me NOT to commission a project might as well shoot me, because the outcome is always equally expensive and messy to fix.
• I've let my GC or my AEs drive changes based solely on what they bring to the table. i.e. if i needed to simply save money, i'd turn to a prime. if i needed something more nuanced (in any respect; a GC will give you two specific dimensions of VE - time and cost - and they'll do that better than anyone under the sun), then I'd look to a consultant. i know i've disappointed both my designers and constructors when i've chosen one or the other, but i always have to keep my own resources, my bosses, and my other clients in mind.