Let me formulate my remark as an answer (and make it more precise):
My experience and knowledge is that to use collocated grids with any standard (lower or higher order) methods for this problem, you must modify (stabilize) your Navier-Stokes equation. it means with some additional effort. This needs not to be necessary when using staggered grids.
The references and some summary I am familiar with are:
S. Patankar: Numerical Heat Transfer and Fluid Flow. 1980 - see chapter 6 "Calculation of flow field" where the problem of checkerboard pattern of pressure values is mentioned for "collocated" grids (i.e. all unknowns at the same grid points) and where only single remedy is proposed - staggered grids, and no modifications of equations or discretization methods is necessary.
J.H. Ferziger, M. Peric: Computational Methods for Fluid Dynamics. 2002 (3rd edition) - it can be seen as a continuation of the book of Patankar. See Chapter 7 on Solution of the Navier-Stokes equation and sections 7.2 on Choice of Variable Arrangement on the Gird, and sections afterwards.
It is noted there that between 1960 and 1980 after introducing staggered grids very few works used the collocated grids. The collocated grids began to be again popular with "projection methods" that exist in many variants and that are well discussed in the book of Ferziger and Peric. If I simplify it the basic idea is to derive and use a Poisson equation to compute the pressure and to update afterwards the velocity field to be conservative.
An introduction of stabilizations (sometimes compared to artificial compressibility) on collocated grids is known to me from papers like:
Schneider, G.E., Raw, M.J.: Control volume finite-element method
for heat transfer and fluid flow using collocated variables — 1. Computational
procedure. Numer. Heat Transf. 11, 363–390 (1987)
Karimian, S.M., Schneider, G.E.: Pressure-based control-volume
finite element method for flow at all speeds. AIAA J. 33(9), 1611–
Nägele, S., Wittum, G.: On the influence of different stabilisation
methods for the incompressible Navier–Stokes equations. J. Comput.
Phys. 224, 100–116 (2007)
So my quite subjective summary might be that when using staggered grids, no additional modification to some "standard" numerical methods is required. Opposite to that for collocated grids some additional efforts is necessary.