Greetings.
I am currently once again attempting to create a service area and while it has succeed in building and importing the points, there is a small issue. While some points generate "properly looking" service areas, others simply do not expand beyond one street while it is connected to the wider road network, like in a photo I managed to provide. I can't figure out what is the issue which prevents the service area from expanding. I have checked the lines and they are all connected to each other, no gaps in here.
I'm not sure I entirely understand your picture provided, but it looks like you have a gap after that initial T-junction. The polygon looks like a buffer of that roadline which stops just after the junction.
I know you've said you don't but, try to run a route analysis to make sure if you haven't done so already.
Also check the Polygon Detail level in your analysis options and possibly increase it to High Precision for troubleshooting purposes.