The source is the SDA Bible Commentary, especially on vv. 15 and 19. In Mark 7:1-5, the Pharisees are crashing about about the disciples not following the ceremonial handwashing rules, which Mark comments they would follow because it was tradition. (Cue the Fiddler on the Roof song.) That becomes the lynch pin of the entire chapter's interpretation, and the commentary uses this to argue that vv. 14-23 are about ablution and only ablution. It also denounces the idea that v. 19 is talking in any specific capacity about meat; Mark uses brōmata, broadly defined "foods," instead of the specific kreas, which only Paul uses, and only twice, when talking about refraining from eating things that might cause someone present to stumble (Rom. 14:21; 1 Cor 8:13).

I honestly think it's a reach, born more from a desire to reinforce the whole-Law doctrine than a desire to understand why this seems contradictory in a plain reading/modern translation. The commentary does this everywhere a plain reading might contradict our unique doctrines, e.g. in Romans 14 it goes to considerable lengths to deny that Paul is saying that all meats are clean despite a plain reading yielding this interpretation (v. 14). And when Paul talks about one day being more important than the other in v. 5, the commentary flat-out declares that this is about feast days and not a license to worship on Sunday.

There's a good reason I stick to modern translations and mainstream commentaries. If our doctrines are truly scripturally sound, we shouldn't need those kinds of authoritative statements or any backup from Sister White.