What relation does ergodicity have to the multiplicity of eigenvalue 1 in Markov matrices

Question

We know that positive semi-definite Markov matrices have eigenvalues $\lambda\leq |1|$ and if the multiplicity of $|\lambda| = 1$ is 1, after raising the matrix to a high power, the matrix will converge to a matrix where each row is the eigenvector associated with the eigenvalue. This is because $M^kv = c_1\lambda_1^kx_1 + ... + c_n\lambda_n^kx_n$ if we write v in terms of eigenbases $v_i = \sum c_i\lambda_ix_i$, and eigenvalues with magnitude lower than 1 converge to 0.
I am interested in why ergodicity relates to eigenvalues? What about aperiodicity and irreducibility points to Markov matrices having just one eigenvalue of magnitude 1?
I feel like I have an intuitive understanding of why both are required for a limiting distribution to exist: irreducibility allows the chain to not get "stuck" in different specific states, so then the stationary distribution must be unique, and aperiodicity disallows oscillation, allowing the stationary distribution to exist, so then we have a single stationary distribution the matrix converges to over time, which is the limiting distribution. And separately, having just one eigenvalue of magnitude 1 essentially implies we have just one steady state that does not oscillate, allowing convergence.
But I have absolutely no intuition on the math that ties these two perspectives on the existence of the limiting distribution together...