Rats are very intelligent animals. Pre-mutation Splinter crudely mimicking his master's moves is a believable novelty for the series' tone imo. And after mutation he's got juiced up intelligence to practice them properly. It's never seemed too outlandish for me. >_>
But really the reason I like the pet origin is because it makes Shredder feel more threatening. The Yoshi/Tang Shen/Saki story is always the same, but Splinter being a normal rat during those events makes Shredder feel like a grand, multi-generational villain. And his reaction when first confronted with Splinter post-mutation, like "Wait? Yoshi's fucking rat? Are you serious?" The dynamics in play are just more interesting. <_<