I know that spectra are supposed to be designed so that suspension is invertible up to homotopy, but I'm having trouble articulating exactly why this is the case.
If $E$ is a spectrum and $\Sigma E$ is the spectrum where $(\Sigma E)_n = E_{n+1}$, then there is a map $E \to \Sigma E$ where $E_n \to \Sigma E_n \to E_{n+1} = (\Sigma E)_n$ is defined using the structure map. I can think of two ways to proceed.
Option 1: Find a map $\Sigma E \to E$, i.e. a map $E_{n+1}\to E_n$ (or some suspension of this). I don't really know how to get a map that decreases degree, though; the adjoint of $\Sigma E_n \to E_{n+1}$ is $E_n \to \Omega E_{n+1}$ which doesn't help.
Option 2: Show that $E \to \Sigma E$ is a weak equivalence by showing that $\pi_r E \to \pi_r \Sigma E$ is an isomorphism. This actually seems wrong: if $E$ is an $\Omega$-spectrum then $\pi_{n+r}E_n = \pi_{n+1+r}E_{n+1}$ for large $n$, so $$\pi_r E = \lim_n \pi_{n+r}E_n = \lim_n \pi_{n+r+1}E_{n+1} = \lim_n \pi_{n+r+1}(\Sigma E)_n = \pi_{r+1}(\Sigma E).$$