Good, good
But a little correction here...
Actually, Roger's illness was revealed four years before he got executed. Once they knew this, they decided to take Crocus at the twin cape of Redline with them along the journey to aid Roger on prolonging his life while finding cure for Roger and Rumbar Pirates (Crocus's deal in exchange of him following the Roger pirate crew voyage as the doctor). Within 3 years, Roger was entitled as Pirate King, and due to the low chance of him to be cured, he seemed to see how it would end and decided to disband the crew.
One year later, he surrender to the Marines. Ofc just as you've stated, the Marines fabricated the real event and said they caught Roger themselves.