At least in one dimensional case for linear advection I have not experienced for finite difference methods the CFL restriction depending on the precision order as you write. Conditionally stable explicit in time well designed numerical schemes have typically the restriction $dt \le C h$, I have such experience (and proofs) with 2nd and 3rd order accurate schemes, the stencil being at most 5 nodes, i. e. from i-2 to i+2.
Opposite to DG methods where degrees of freedom for higher methods are localized at the same geometric location (element), the standard finite difference methods are increasing also the geometric stencil when increasing the precision order.
Standard text books on finite difference (or related finite volume) methods relate the CFL condition to a "domain of influence" that depends on the equation you solve. Maybe some "geometric" clustering of degrees of freedom in DG method introduce the unfavourable influence of CFL condition on the order of method, but my subjective opinion would not expect it with finite difference methods.
I was asked for references, here is one:
Principles of Computational Fluid Dynamics - Pieter Wesseling, 2001, see Theorem 9.3.3 on so called kappa class of schemes for 1D advection equation $u_t + v u_x=0$ with constant speed $v$. Several 2nd order schemes belong to this class and one 3rd order accurate (QUICKEST), the stability is reached for Courant number $dt v/h$ less or equal 1. It is so called von Neumann stability (Fourier series analysis).
I do not work with higher order methods than 3, but I know from papers and conferences that e.g. so called ADER schemes (M. Dumbser use it also for DG) can produce arbitrary order of accuracy and I do not know about any bad behaviour of schemes with respect to the CFL condition, but I might be wrong.