P Basses aren't favored by engineers because of speed - that's all up to the player. The reason engineers like them is that they put out a lot of mids. That way if, say, the kick drum is sounding sweet at 300 hz, you can duck the bass there and still have a lot 200 hz and 400 hz to stand up in the mix. In the same token, if you've got a Jazz bass with both pickups dimed and a relatively narrow band of mids in the same spot that kick drum sounds sweet, it will take a lot more work to get the bass to sit well in the mix (or you compromise on the kick). I'm not a huge P Bass enthusiast by any means (I tend to like double buckers) and still I own 3...Honestly, prioritizing speed at all in the studio is usually going to be counterproductive. People get tense and don't play as well or listen as closely. Numerous people have lauded Ps as "fast" to record. To me that's first of all false, but secondly it's about the last thing you should really be worried about it. "Does it sound good for the track?" should be number one and anything else is a distant second.
What you really want is to work EFFICIENTLY but not hurriedly. There's nothing inefficient about a MM or any number of non-Fender basses if that's what you're going for. There's nothing mythical about the P bass. It's a great all purpose bass and they were the default for a lot of stuff because they were the first/only. But not we have many many many many many many more options to use. So we should use them. If P's were so perfect, why don't we just burn all the rest and have everyone play a P? How many of you would sign up for that? How many of the "oh, just always use a P to record" people play something COMPLETELY different the whole rest of the time. Why wouldn't you just record with THAT bass? Maybe it's because I think being a joiner is stupid but to me just recording everything with a P goes against everything great about music. It's all those little variation and changes that make things interesting. Why did Queen use a Wurlitzer instead of the much "better" Rhodes on Youre My Best Friend? I dont know but THAT'S the sound of the track. Meanwhile, Pink Floyd's Sheep would not be the same with a Wurli instead of the Rhodes. It's all about finding the right tools for the job. Sometimes that will be a P, but it should never be assumed that it will be just because it's been around the longest. That's just pure laziness.
So my "final answer" is "whatever the bassist feels most comfortable with and inspired by and sounds good in the track".