The best way to start as an independent sponsor is when your (PE or FO) employer doesn't want to proceed with a deal that you love, and then you quit to sponsor it independently.
This way you've already sourced the deal on someone else's dime, and you have formed a relationship with the seller. The uncertainty, duration, and cost of a search is already behind you; now you just have to close the deal.
Listen to the latest episode of the Minds Capital Podcast episode to hear that this is exactly what Steven McGrath did his employer decided to discontinue PE direct deals. He quit his job, persuaded the seller to re-sign the LOI with him directly, and closed the deal shortly after.