The beneficial bacteria we need are not planktonic, they are sessile, (they live as biofilm on surfaces), on media like ceramic rings, lava rock, or bio balls, even tank glass. They are not floating around in the water and helping to consume ammonia in any real sense in a planktonic state.
So using old water is basically futile. And in fact, using it can cover up what's really going on in a new cycle.
This is because old tank water already is most likely partially or even highly saturated with nitrate, so it can give an aquarist the false sense of security that the cycle is complete, when those nitrates are left over from the old water, and ammonia and nitrite are not being consumed.
One is better off using new water, so you watch the progeession of the cycle from ammonia, to nitrite, and finally nitrate, if starting a cycle from scatch .
And when adding media from a seasoned filter, to keep it seasoned, those bacteria need to be fed with ammonia, so adding "a few fish" to a tank with a seasoned filter is advised, if fish aren't added the bacteria in the seasoned media could starve.