Yes, parsing HTML is doable. But it must be done manually - annoying if you have HTML within the TD’s. The CSV export does all of this already: eliminating HTML tags, doing the separators and masking for CSV, etc. And if the number of columns or the table formatting changes, you can run into problems.
But as @ibuchanan mentioned, the table filter wrapper including the CSV export obviously comes with an extension and is not a build in feature. Did not know that - my bad. If it’s not build in, then it’s very unlikely that there is a common CSV export function / download URL / API call. Sorry for the confusion.