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Using the formalism of Ando-Blumberg-Gepner-Hopkins-Rezk we can identify $$\mathcal{M}=\Sigma^\infty_+ EGL_1(S_p) \wedge^\mathbb{L}_{\Sigma^\infty_+ GL_1(S_p)} S_p.$$ This smash product only involves connective spectra, so $$\pi_0 \mathcal{M} \cong \pi_0(\Sigma^\infty_+ EGL_1(S_p)) \otimes_{\pi_0(\Sigma^\infty_+ GL_1(S_p))} \pi_0S_p\cong \mathbb{Z}\otimes_{\mathbb{Z}[\mathbb{Z}_p^\times]}\mathbb{Z}_p.$$ You can see this by examining the $\mathrm{Tor}$ spectral sequence.