Persistence barcodes and spectral sequences

The answer to your question is no, nobody has used persistence to improve the algorithmic efficiency of computing differentials, although of course the relationship between persistence intervals of a filtration and various terms in its Leray spectral sequence have been described rather explicitly by Basu and Parida.

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.

there is actually a bit more to this story. The computational topology community has indeed wrestled with spectral sequences. The connection to spectral sequences has been discussed since as early as the Carlsson & Zomorodian paper mentioned by Vidit (although, not necessarily explicitly in that paper).

To make the connection clear, the $E_\infty$ page is a graded $k[t]$-module containing the "infinite" bars. the bars of length $p$ aren't exactly on the $p^{th}$ page, but, when you compute homology on page $p-1$, you can find the bars of length $p$ by hanging onto the images of the $p$-th differential.

Speaking informally, in the language of numerical linear algebra, this is like looking at successively larger block diagonals of the boundary matrix written down in an appropriate way, reducing them, and then ``expanding."

Infact, this algorithm (albeit not the direct connection to spectral sequences) is written down in the paper ``Clear and Compress" due to Bauer et. al

In particular, there is also the question of whether other spectral sequences speed up computation in a practical sense. There are results that suggest asymptotic improvements in certain cases. In practice the jury is still out.