St. Matthew's vs. St. Matthew

This may depend on factors other than a consideration of traditional grammar.

There is a St Matthew Church (not St Matthew's Church) in Charlotte, another in Glendale Heights, another in San Mateo ...

Thus one finds:

Rev. Evans and all of us at St. Matthew welcome you!

[Internet]

Though St Matthew welcomes you probably needs rephrasing using notional concord!

This is in line with the growing trend to drop apostrophes in constructions where association rather than true possessives are involved:

Achilles tendon (still sometimes apostrophised)

Pilton Working Mens Club

Waterstones Bookshops

childrens clothing department

St James Park (amongst others), etc ...

although these examples all retain the s after the apostrophe has been ousted, which sometimes produces frightening new words like mens and childrens.

It's going to be St. Matthew's, because you can read it as either:

The Church of Saint Matthew

or

Saint Matthew's Church

If you shorten it, only St. Matthew's makes sense.