Your reasoning is correct: **opening the live wire stops the normal load current in both live and neutral.** A single-pole switch is therefore sufficient for ordinary on/off control in a correctly wired circuit.

The advantage of double-pole switching is **disconnecting the appliance from both supply conductors**. This matters because a wire can carry no current and still have a dangerous voltage relative to earth.

Two situations explain the benefit:

- **Live and neutral are reversed.** A single-pole switch could then open the actual neutral. The appliance stops operating, but its internal wiring remains connected to live. Touching that wiring while grounded can complete a circuit through your body. A double-pole switch opens both conductors.
- **The neutral develops a dangerous voltage.** Neutral is normally close to earth potential, but supply faults or wiring errors can raise its voltage. Opening both conductors also disconnects the appliance from that neutral. Electrical Safety First specifically cautions about working on neutral conductors where single-pole isolation is used. [Safe isolation guidance](https://www.electricalsafetyfirst.org.uk/media/1201/best-practice-guide-2-issue-3.pdf)

So **a single-pole switch can serve as an operating switch; a suitable double-pole isolator provides more complete disconnection for safety**. Whether single-pole isolation is permitted depends on the supply arrangement and applicable electrical rules.

Before working on equipment, the supply must be safely isolated and the relevant conductors proved dead—even with a double-pole device. [HSE safe working guidance](https://www.hse.gov.uk/pubns/priced/hsg85.pdf)