If by $\pi_1^{st}X$ you mean the unreduced version (as you must when you say that $\pi_1^{st}(pt)=\mathbb Z/2$), then for a based space $X$ you can functorially split off $\pi_1^{st}(pt)$ from $\pi_1^{st}X$, so there is always that kernel.
If you mean the reduced version, then you're precisely talking about the other factor in that splitting.
If we define $\pi_k^{st}X$ as the direct limit of $\pi_{k+n}(\Sigma^n X)$, then we get the reduced version. (Your $\mathbb Z/2$ may be considered as the unreduced $\pi_1^{st}$ of a point, or as the reduced $\pi_1^{st}$ of a $0$-sphere.)
As for the spectral sequence: You probably mean the one that goes from $H_i(X;h_j(X))$ to $h_{i+j}(X)$ (everything unreduced) for any generalized homology theory. Here $X$ is not necessarily based or connected. If $X$ is connected then, choosing a basepoint, you can split off the left ($i=0$) column, so to speak. Or work with the analogous spectral sequence that goes from $H_i(X,pt;h_j(X))$ to $h_{i+j}(X,pt)$ and lacks that column.
That applies to any $h$. By looking separately at all components (or by choosing one point in each component in each component and then noting that this discrete set is a retract of $X$) you see that in general the canonical map $H_0(X;\pi_j(pt))\to h_j(X)$ is a naturally split injection.