For a governess in the Georgian era, Miss Taylor was hardly poor Miss Taylor but most fortunate in both, a good position with her employer Mr Woodhouse and being treated well as part of the family. Then, after her pupil Emma is older, being able to marry a kindly man of some fortune and have her own home. Mrs Weston's combined luck of position and later marriage & establishment was not typical of the Georgian era governess.
Typically, a Georgian era governess received about £25 per annum from which she had to save for her retirement; it would be very difficult for any governess to be independent in old age.
The position of the governess was not an enviable one; insecurity and reliance on your employer and their family, poor health would've made the position of governess even more difficult for a young lady such as Jane Fairfax. The lot of a governess largely depended on the family who you worked for.
Some governesses in JA do appear to stay on with their employers; Mrs Jenkinson who had been Anne de Bough's governess remained as her companion. Mrs Younge who had been employed by Mr Darcy as Georgiana's governess, became her companion until Ramsgate.

The position of the governess in JA's time has been on the L&T board before and if you trawl through the L&T discussions using key terms such as teachers, governess you'll find many discussion threads and informative posts of former L&T host, JulieW.

Below, I've included a past discussion on the governess to start you off !
In the posts, follow through links to find more discussion on the Georgian era governess.
As further reading, check out Myretta's post for a recommended book; 'A Governess in the Age of Jane Austen'; Miss Agnes Porter as a governess moved around several families in very early C19th. I read this book ages ago, and recall Miss Porter's letters made clear even a lady determined to make the best of her role as governess could feel isolated and insecure in her tenure.