> Blk/Wht Brn/Wht
There's a reason those wires have "Wht" on them.
In house-wiring you always go White to White. (The few exceptions are supposed to be re-Identified non-White.)
White is *supposed* to be the less-live side of the wall-wiring. (Often measures 1V-2V to Ground, the other wire is ~~120V to Ground.)
So they *may* have wound that iron so the White-stripe wires have less insulation than the non-White (Black, Brown) wires, which would have the best insulation.
OTOH, "white to white always" is widely ignored in house-wiring. 60% of the outlets in my old kitchen had White/Black reversed. You can't trust wall outlet hot-side. So they have to put dang-good insulation on both (all 4) winding-ends. Your way "will work".
However, given a White, it is probably best practice to connect it (them) to the white side of the line-cord.
In house-wiring, you NEVER (since 1915) switch or fuse the White wire (except with a DP which also breaks non-white). INside a plug-connected chassis, this was historically not-observed. Finding a switch or fuse in the "neutral" side was common, and "safe" because you should always UN-plug before going inside. However newer safety codes are starting to require all switch/fuse in the non-white side, so this is becoming best practice. (But don't chop-up your 1953 classic just cause the fuse is "in the wrong side"!)