A toner won’t change the colour of your roots, no.
Now you’ve put up another picture, I agree it‘s not a great job and could definitely be improved.
The original pics you posted were only showing as thumbnails for me so they didn’t look that bad.
However, whether the current hairdresser can improve it or not, I can’t really say. The fact they’ve suggested just adding a toner sounds like they’re not too interested in whether the client leaves happy or not.
Have you any friends you can ask for a recommendation for a good colourist for where you live? Years ago there was a young woman with fabulous hair working on the supermarket checkout and I complimented her hair and asked her where she got it done. She was happy to tell me and that was my hairdresser sorted for the next few years.