It depends entirely on the journey being made. Most of the time it seems like, yes, journeys from Walsall to stations on the line between Kidderminster, Smethwick Galton Bridge, B'ham Snow Hill through to Leamington/Stratford can be made via Smethwick Galton Bridge.
Walsall to Stourbridge Jn, for example, is permitted via Perry Barr and Smethwick Galton Bridge.
Walsall to Stratford-upon-Avon is also valid changing at Smethwick Galton Bridge, even though you'll go through Birmingham New Street twice to do so.
There are likely some journeys which cannot be made this way, though. I'd certainly recommend doing a search via Smethwick Galton Bridge to make sure it's permitted for any particular journey you choose to make. For example, you cannot travel to Leamington Spa by changing at Smethwick without purchasing a combination of tickets for your journey (but you can to Warwick, strangely).
The forum's ticketing site will allow you to explicitly choose to "change at" Smethwick Galton Bridge, and also prevent changes at stations like Birmingham New Street and entirely disable walks between stations to help you get the exact itinerary you would like.