The answer to your question is no, nobody has done this yet, although it is an interesting idea and might be taken up in the future. In fact the flow of ideas has gone in the opposite direction, with attempts being made to use spectral sequences for parallelizing persistence computations (see here for one example).
Here's an elementary observation: as mentioned in that article by Carlsson and Zomorodian, every sequence of $k$-modules admits a straightforward reinterpretation as a graded $k[t]$-module where $t$ acts by moving things forward one step along the grading. The existence of a persistence barcode relies crucially on the structure theorem for graded modules over graded PIDs. Since $k[t]$ is a PID only when $k$ is a field, relying on persistence will not solve any extension problems for you when you try to compute differentials -- all your $E_{\bullet,\bullet}^\bullet$s will have to be vector spaces already.