To me, the flight looks fine at first, and the wiggling only starts later. Skip forward to about 1:20 in the video, when the slow-motion replay starts. Right off the rod, the rocket is as straight as a lamp-post. At 1:22, it tips slightly to the left but is still going pretty straight. At 1:24 it swerves back to the right and by 1:25 there's a visible zig-zag in the smoke trail. The rocket continues to zig-zag at 1:28 and beyond. Also worth checking is the pad camera footage starting at about 1:37 and repeated in slow motion at about 1:48. The rod does wobble a bit but the rocket flies straight; it does not start zig-zagging until it is well clear of the rod.
The rocket appears to be a Mercury Redstone, with small fins and probably a heavy nose. That combination is indeed prone to zig-zag a bit once it reaches high velocity. The small fins don't produce much correcting effect until the rocket is significantly off course, and then they produce a lot. The rocket swings back into line, swings past in line, and repeats the process in the other direction.
A longer rod may make the rocket more resistant to weather-cocking, but since the effect described above happens at high velocity, a longer rod won't have much effect on that. Bigger fins will, but then it won't be a scale Mercury Redstone any more. I'd say that, although it isn't perfectly straight, that flight is safe enough in that the rocket goes near enough straight up - it's not as if the rocket tips hard over shortly after launch and goes on a low trajectory.