No, a 4001/3 does not sound like a P bass or a J bass, nor is it supposed too, but it seems many like to compare the three. It's like the same thread about tail lift being a problem that comes up every now an again. Rickenbacker basses have not had tail lift issues in a dogs age but that does not stop the topic from coming up.
If you are looking for serious thump, I suggest a 4004. That bass looks like a Rickenbacker but it's a low end and all around monster! A J bass? Try a 4002, you'll never look at your J bass the same way again. Both these basses have very powerful humbucking pups and they are configured differently than the 4001/3.
Yes, 4001/3's have more kick in the low mid low range and that is exactly where I prefer to be. Rickenbacker basses work for me and my sound. They are not for every one and that's a good thing. And this is coming from a guy who "cut his teeth" with fender P basses in my early days.
Sepp