3s may use 6s' qualities to cover up for their personal/selfish agenda. Projects are allegedly done to benefit the whole, whereas the "whole" often gets narrowed down to whoever is closer to the top of the hierarchy (whereto the resources accumulate). Businessmen are said to enter politics for further upward mobility and political or business leaders are also noted for disappearing after having been unable to carry the weight of the responsibilities coming with their position/ fulfill their promises and/or having been exposed to some corruption scandal. I'd say, aesthetics-wise, the 3 playing the 6 is like folk-dance rehearsed in a museum with perfect costumes and a group of classically trained professional musicians. There will be some pretense there, extra smooth/slick cleanliness to a stereotypically poor man's "dress code."
I think a lot of the 3s at types as ISTJ, ESTJ, ESTP are actually core 8s. With secondary being 6 in their tritype. They aren't "heart" oriented, aren't really emotional but their greed, lust, usurpation, will for power and owning material stuff may manifest as a "progressive" attitude in terms of aesthetics. They may want "the best," like a sports car, a smart watch, a state of the art house, biz suit, items that dress them with fluidity, smooth out their inner aggression into the form of competitiveness. To make it look like their anti-social (E8), non-communitarian drives have validity to them, that they are the path-blazers of the future. These are the more blatant 3s, I guess, who are actually 8s. Core-fix 3s, like partially myself, I think may appear more 6-like, perhaps at ISTJ, INTJ, more cerebral 5-like. Here 3 is more emotion-based, I'd say, and the gut type, that supported the tertiary 3s comes later. In my case at least I'm more narcissistic with almost the same aesthetic tastes, although less ambitious or driven to achieve. Primarily I'm more image driven, although since with less effort it almost always fails, the aspirational side of my personality tries to go beyond that vanity-drive to look for help in community (6) and realism over aesthetics of image (1). So real 6s will try to hide their fear and real 3s will sublimate their bragging (trying to impress you with their competence) into, say, the form of "help" - the way I'm doing it now. Tertiary 3s will brag more blatantly with their position and taste only to hide their profane, non-exalted/rarified drives. IMO. Core-fix 6s when not afraid will turn to gut based pleasures, Core-fix 3s with a secondary 6 when relaxed will let down their guard to show their narcissism and sense of entitlement.