No it doesn't. It's very difficult to imagine a plausible circumstance where you have a company that took significant investment where you could hold a board seat that the board could not eject you from after you left the company. You can't sell a board seat.
As a hypothetical, kind of? But not really. The board is written into the company's bylaws, as a rule, and requires a board vote to change. 'Selling your board seat' really means engineering a complex deal that requires a bunch of other people to sign off.
The same is true of selling your equity, by the way. As a founder you have common shares, but early-stage investors want preferred shares with QSBS treatment. Even if you're allowed to sell your shares, which most startups don't let you do, it's not in your power to convert them to preferred or give the buyer QSBS treatment.
"Yes, I'll put another hundred grand into the company, but I want a pref issue with an attached board seat."