No. Consider, for example, $M=\mathrm{SL}(3,\mathbb{R})/\mathrm{SO(3)}$ endowed with its $\mathrm{SL}(3,\mathbb{R})$-invariant Riemannian metric $g$ (which is unique up to a positive constant multiple). This is an irreducible symmetric space of noncompact type. The identity component of the isometry group of $(M,g)$ is $\mathrm{SL}(3,\mathbb{R})$ itself (since $\mathrm{SL}(3,\mathbb{R})$ has trivial center), and the exponential map of $\mathrm{SL}(3,\mathbb{R})$ is not surjective.
To see this, note that, when all of the generalized eigenvalues of $a\in{\frak{sl}}(3,\mathbb{R})$ are real, then $A=\exp(a)$ has positive (generalized) eigenvalues, and, when $a$ has one real eigenvalue and two distinct complex conjugate eigenvalues, say, $z\not=\bar z$ and $-(z{+}\bar z)$, the eigenvalues of $A = \exp(a)$ are $\lambda = e^z$, $\bar\lambda = e^{\bar z}$, and $1/(\lambda\bar\lambda) = e^{-z-\bar z}>0$. Thus, $A = \exp(a)$, cannot have eigenvalues $(-\tfrac12, -2, 1)$ since $-\tfrac12$ and $-2$ are not complex conjugates. In particular $A = \mathrm{diag}(-\tfrac12, -2, 1)\in\mathrm{SL}(3,\mathbb{R})$ is not the exponential of anything in ${\frak{sl}}(3,\mathbb{R})$.
You'll run into the same problem with the conformal group in this example, because the space of conformal transformations in this example is the same as the space of isometries.
Added comment in response to OP's edit: Unfortunately, I think that, without more restrictions on the class of Riemannian metrics of interest, this is probably not a very sensible question. For example, for any connected semi-simple group $G$, the generic left-invariant metric on $G$ will have $G$ as the identity component of its isometry group, so there are many examples of such metrics for each $G$ for which the exponential map is not surjective (which happens for very many if not 'most' non-compact semi-simple Lie groups). Thus, for example, the generic left-invariant metric on $G=\mathrm{SL}(2,\mathbb{R})$ gives a $2$-parameter family of $3$-dimensional examples that are distinct up to homothety.
