In general, all partners sell. But in some partner roles, that's almost all they do, whereas in others, they have huge responsibilities in project delivery, apprenticeship (training others), firm leadership etc. So as usual in consulting, "it depends". How? Well, it depends hugely on the type of consulting - principally the degree of standardisation vs customisation of the services offered, and degree of trust/client intimacy involved in the work.
Highly customised/high trust engagements lean towards more senior involvement in *delivery* of the work and "apprenticing" the next generation, as well as the sales process (and low ratio of non-partners to partners - maybe 7:1). e.g. strategy consulting at MBB. These firms are also privately owned by the partners, which means firm leadership roles also fall on the partners.
Conversely highly standardised, lower-trust/intimacy work lends itself to the partner being primarily focused on sales, and having little involvement in execution, which is carried out by a larger number of execution resources per partner (Maybe 15-20:1). e.g. implementation consulting at the Big 4.
David Maister (who is well known as a consultant to consulting firms!) writes about these kinds of structural differences e.g. his book Trusted Advisor. Or this article: https://davidmaister.com/articles/the-anatomy-of-a-consulting-firm/