Should the Independent Sponsor always be the Chair of the Board?
Yes, it sounds nice, you control the narrative, and nobody will expect otherwise.
However, it's a missed opportunity to appoint another leader in the company. You'll have a CEO; why not ask one of the other carefully sourced board directors to be the Chair? You'll be around to add your value regardless.
Outsourcing it to one of the independent board members helps reduce your workload, democratizes the overall narrative, facilitates closer collaboration between the CEO and the other board members, and further enhances the credibility & professionalism of this SPV.
As compensation, you'll have to increase the comp to be a monthly fee ($5-10k) plus significant stock options. A chairperson should be expected to have highly regular contact with the CEO and attend monthly financial reviews, at least.
