The presence of the but in that statement indicates it isn’t true. And that would be the case for any “I support” statement.
To me it seems like the obvious point of prepending a support statement to a separate objection would be to clarify that what you mean by that objection is not broad hostility, if it seems people might confuse it for that otherwise. There's better ways to word it (maybe split into multiple sentences), but I don't think someone saying "but" necessarily is trying to convey that the thing they are objecting to competes with or outweighs their support.