I think the problem is you only have 2 schools, so your global posterior is not strongly conditioned by your data.
What you are suggesting sounds like complete pooling the estimates of the two schools instead of partial pooling like the hierarchical model does. If that’s what you want you can model it like that.
In terms of extracting summary statistics afterwards you can do whatever you want, just keep in mind that your model uncertainty about the global effect may be the plausible thing even if it feels underwhelming.