Heck yes. This is exactly why I roll my eyes every time I see someone suggest to "just roll off the tone knob" on a bass with rounds for a flats sound... just completely misses the mark. The best thing about flats is that fat punch that comes from how the flat outer ribbon winding makes them move. Maybe with super heavy EQ and signal processing you can make rounds get in the flats ball-park, but you're gonna have to do some significant boosting to even get close. Rolling off the tone-knob only cuts treble, might subtly impact the resonant peak, but not in a way that turns a tack hammer into a sledgehammer.
If you understand and appreciate that thwap, you might want to try some GHS Pressurewounds on one of your other basses. They're the only "hybrid" type strings that I've found, that can get close to the "best of both worlds" thing.
As much as I do like flats on a P, my favorite string for my '65 Precision so far has been D'Addario Tapewounds. Typically a tad brighter than flats and a bit more sustain, but they also have their own unique punch that's hard to describe. If the rounds are the tack hammer, flats are the sledgehammer... maybe tapes are the dead-blow hammer? So those might also be worth trying out if you haven't. Much like any other style of string though, all tapes aren't equal either, so lots to explore, but I generally like D'Addarios for the velvety finish, somewhat lower tension, and the gauges are pretty standard, so usually no nut-filing is required.
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