@Jiali
The BTG increases the portal production
boost for any guest race portal.
The Campus is the S&D equivalent of a portal. A lvl 4 campus provides a 60% boost to all
faculty productions (so you are correct, it won't improve apprentices). A BTG at lvl 1 will increase that faculty production boost by 6%. That percentage boost increases as you upgrade the BTG. The Wiki has a table that shows each level and the accompanying percentage boost
here. Sometimes those tables are collapsed; if so, click the 'expand' in the top right of the 'overview' table to see the details.
In Halflings, the Spring Grove is the portal. At lvl 4 it will provide a 60% boost to all farms and fields productions. Again, the BTG will increase that production boost by 6%+ depending on its lvl.
The BTG impacts the production of all guest race goods that are boosted by higher levels of the portal, not just the goods with capped storage like the PP's do. That impact is larger the higher you upgrade the BTG. Whether that impact is worth the loss of the MH, only you can decide.