Bluntly speaking, SUPG and alike and RANS are different approaches to different problems that, however, have the same name - instability - and the same phenomenology - the failure of numerical routines.
RANS is used to cope with turbulence as an instability of the equation. If a flow is or becomes turbulent the describing equations are instable, e.g. because of bifurcation. This means that a small disturbance may lead to large changes in the solution. I tend to see RANS as a reformulation of the problem, so that the equations are stable and numerical algorithms are applicable.
SUPG and alike eliminate instabilities of numerical algorithms.
Regarding your question whether you can use SUPG instead of RANS. I would say no. If your problem is unstable, also stable numerical algorithm are not convergent. Your algorithm may converge to a solution, but you cannot be sure that you are on the right way.
The common point I see is that turbulence occurs at high Reynolds numbers, where also the factor space discretization times velocity magnitude $hv$ tends to be large, i.e. where upwind stabilization, as it is mimicked in SUPG, is necessary.