The Leitstrahl system you're talking about was a late development which was fitted to a small number of V-2s starting in Decembmer 1944, with only one V-2 battery (SS 500) actually having the guidance equipment. With or without this system, the V-2 was only guided during its initial powered flight of approximately 30 seconds. The guidance system of the V-2 enabled it to launch vertically, make a controlled tip-over maneuver onto the desired target heading (which was either pre-programmed or indicated by the radio guidance beam), and then turn off its rocket motor when it reached the correct velocity. Once motor cutoff occurred, the rocket had no further ability to make any course correction or adjustment.
Even with the addition of the Leitstrahl system, the V-2 was only "accurate" to within a margin of ten kilometers or so, sufficient to generally hit somewhere within a targeted city but utterly incapable of aiming at anything specific within that city. The *one and only* time V-2s were aimed at a specific tactical target, the Ludendorff Bridge, eleven Leitstrahl-fitted V-2s were fired by the "elite" SS V-2 battery . . . and not a single rocket came within miles of the bridge. Hitler jubilantly ignored this reality and congratulated the V-2s for the bridge's destruction.
In the end, the V-2 program was an absolute failure, costing over 2 billion Reichmarks and doing little meaningful damage to Britain.
(Note that Hitler, with his usual abject stupidity, also literally failed to comprehend that the radio guidance was an upgrade, and publicly bragged that the elimination of the radio guidance made the V-2 even more lethal by making it impossible for the British to jam the guidance. )