I think there is an underlying assumption that the answer will not depend depend on which definition of "simple" is involved. In principle, a group is simple if it has no proper normal subgroup. However, since Lie groups are related to Lie algebras, we can also bring in the notion of a simple Lie algebra, which is one having no proper ideal. (For the algebras, this can be further generalized to the definition of a semisimple algebra, which has no nonzero Abelian ideal. A semisimple alegebra is then a direct sum of simple algebras.)
Sometimes, the notion of a simple Lie group means one that has a simple Lie algebra. However, this conflicts with the strict group theoretic definition—although the conflict is not actually as bad as it might look. In a small enough neighborhood of the origin, the Lie group and the Lie algebra are in one-to-one correspondence, which means that if the algebra is simple, the group also looks simple in the immediate vicinity of the origin.
This has a lot of important consequences, the key one being that it is always possible to form a quotient of a Lie group with a simple Lie algebra by a discrete group to get another Lie group with the same Lie algebra. That is, if $G_{1}$ is a Lie group whose Lie algebra $g_{1}$ is simple, the there is a discrete normal subgroup $Z$ of $G_{1}$ such that $G_{2}=G_{1}/Z$ is simple, and the Lie algebra $g_{2}$ of $G_{2}$ is isomorphic to $g_{1}$. One of the best known examples of this is that the special orthogonal group $SO(3)$ and the special unitary group have the same Lie algebra, $so(3)\cong su(2)$, and the two groups are related by the quotient $SO(3)=SU(2)/\{\pm I\}$, where $\{\pm I\}$ is the $C_{2}$ group consisting of the positive and negative $3\times3$ identity matrices.
This means, for example, that the perturbative quantum field theory with a gauge group $G_{1}$ looks almost exactly the same as the perturbative theory with the gauge group $G_{2}$.* There is more to the mass gap problem than just perturbative physics, but it is generally believed that there can be no essential difference between the behavior of Yang-Mills theories with $G_{1}$ and $G_{2}$ (or any other intermediate gauge group). In fact, it is presumably felt that the equivalence will be trivial to demonstrate, once the structure of the Yang-Mills theory is well enough understood to demonstrate the presence of a mass gap (and/or color confinement) rigorously. In fact, it seems like it ought to be very simple to generalize from a simple compact gauge group to any compact Lie group.
*The only difference is that some representations of the larger group $G_{1}$ may not correspond to representations of the smaller $G_{2}$. In the example with $SO(3)=SU(2)/\{\pm I\}$, the larger group $SU(2)$ has unitary representations of every positive integer dimension, but $SO(3)$ only has unitary representations with odd dimensions. [This is the well-known fact that the Lie algebra $so(3)$ with commutation relations $[J_{i},J_{j}]=i\hbar\epsilon_{ijk}J_{k}$ only has orbital angular momentum representations with integer angular momentum $j$—and thus odd dimension $2j+1$.] However, this difference in the representation structure only affects how the gauge field can couple to charged matter fields. It does not affect the pure Yang-Mills theory that is the subject of the Clay Millennium Problem, since that involves only the adjoint representation of the gauge group; and since the adjoint is defined by the action of the group on its own Lie algebra, it exists with the same essential structure for both $G_{1}$ and $G_{2}$.