So, instead of introducing Jane to marriage partners, her gentle friends insist that she apply herself for work as a governess, persumably because it is one of the few professions that will allow her to retain her status as a gentlewoman

They dont "INSIST". Jane is the one who insists that she's going to start work as a governess. The Campbells do introduce jane to society but problaby because she has no dowry, she does not get any offers. She is nearly 21 and decides that since she's got to earn her living somehow, she needs to start...
and yes naturally since she does have to earn a living her friends wish her to engage in a profession which will allwo her to keep her status as a lady and possibly to meet occasionally with gentlemen. Why shoudl tehy wish for her to take up soem job which would remove her from her social status?
I dont know waht you mean about "Cox".. I dont see taht William Cox is likely to propose to Jane?