I suppose it's reasonable that the transporter draws power from the same circuit as the engines
In TOS, perhaps - but the other shows suggest the transporter actually consumes relatively little power (it's once powered by a single hand phaser, whereas a shuttlecraft needs to drain several of those to lift off) and can keep on working when many principal systems are down.
Since Scotty speaks of velocity balance variation, and probably isn't referring to the velocity of the ship (since they aren't going anywhere) but to transporters (because his report is all about repairing those), blssdwlf
's idea that these wires transfer excess momentum between the transporter and the engines is a pretty fitting one.
Still, it's a bit of cabling. Scotty could bypass it in a heartbeat, tearing out matching wiring from the nearest piece of less vital machinery. So I'd argue that the damage to the wiring was more or less irrelevant, and so thus is its placement, too. What matters is that firing a phaser at those cables fried the ionizer (by sending a spike, or by cutting vital power that kept the ionizer from being fried), a machine that is located somewhere else altogether.