I've successfully edited the field aliases for a hosted feature service. However the changes do not seem to be propagating through to web maps that are using the service, even if I set a refresh for the layer and clear my browser cache. If I add the layer in again to a web map the aliases are displayed correctly in the attribute table and when you configure the pop-ups.
So how do I get the aliases to show in existing web maps without having to add and reconfigure the layers?
Where did you update the alias' and how did you publish your service? Is it a hosted service? Did you update the alias' in the layer, on the service or in the web map?
I'm not sure if you're still looking for a solution but this worked for me.
I had the same issue as you, it looks like on a web map the "Alias" field only works for pop-ups. If you want to change the name of a field when people look at the data in a table format instead of pop-up, you need to change the "Display Field".
Set Display Field section on this page worked