Not stupid at all. Definitely yes. Don't have the numbers on hand but it's orders of magnitude more CO2-equivalent released per kg-mile, especially when you factor in the fact that they are using methane.
Of course the reality is that this tech won't ever see adoption used that widely, but where is the break-even point?
Full flow staged combustion engines like Starships do not have significant un-burnt methane. They run slightly fuel-rich, but that results in extra CO emissions rather than CH4 due to the temperatures involve -- methane cracks at 1200C, Starship engine temperature is 3000C.
Starship's operations in Boca Chica do emit methane during ground operations. The mitigation for that is to use a pipeline rather than trucks for delivery.
Solid rocket motors emit all sorts of nasty stuff, like aluminum particles.
Of course the reality is that this tech won't ever see adoption used that widely, but where is the break-even point?