I am no expert in the general theory, but let me share some thoughts. In some sense (c) accounts for the fact that $K$ does not have enough open subgroups (unlike at nonarchimedan places), so that the usual convolution definition of Hecke operators is too crude. Instead, one needs to work with the convolution algebra of all distributions on $G$ with support in $K$, i.e. with $U(\mathfrak{g})\otimes_{U(\mathfrak{k})}A_K$, where $\mathfrak{k}$ is the Lie algebra of $K$ and $A_K$ is the convolution algebra of finite measures on $K$. For compatibility with (b) one needs to restrict to elements of $U(\mathfrak{g})$ which commute with elements of $\mathfrak{k}$, and a simple way to achieve this is to restrict to $Z(U(\mathfrak{g}))$. In this way one can obtain a unified treatment at all places for which the adelic framework is most suitable. For example, a Hecke cusp form on $\mathrm{GL}_n$ can be characterized by $n$ parameters at each place: these parameters define the local $L$-functions whose product is the global $L$-function.

For a special case see Section 2.3 in Goldfeld: Automorphic forms and $L$-function for the group $GL(n,\mathbb{R})$. For a brief discussion of the general theory see Borel-Jacquet: Automorphic forms and automorphic representations, Proc. Symp. Pure Math. 33 (1979), 189-202.

I am no expert in the general theory, but let me share some thoughts. In some sense (c) accounts for the fact that $K$ does not have enough open subgroups (unlike at nonarchimedan places), so that the usual convolution definition of Hecke operators is too crude. Instead, one needs to work with the convolution algebra of all distributions on $G$ with support in $K$, i.e. with $U(\mathfrak{g})\otimes_{U(\mathfrak{k})}A_K$, where $\mathfrak{k}$ is the Lie algebra of $K$ and $A_K$ is the convolution algebra of finite measures on $K$. For compatibility with (b) one needs to restrict to elements of $U(\mathfrak{g})$ which commute with elements of $\mathfrak{k}$, and a simple way to achieve this is to restrict to $Z(U(\mathfrak{g}))$. In this way one can obtain a unified treatment at all places for which the adelic framework is most suitable. For example, a Hecke cusp form on $\mathrm{GL}_n$ can be characterized by $n$ parameters at each place: these parameters define the local $L$-functions whose product is the global $L$-function.

For a special case see Section 2.3 in Goldfeld: Automorphic forms and $L$-functions for the group $GL(n,\mathbb{R})$. For a brief discussion of the general theory see Borel-Jacquet: Automorphic forms and automorphic representations, Proc. Symp. Pure Math. 33 (1979), 189-202.