light fixture wiring

In this old house and renovated probably in the 70s, I was changing a ceiling light, but something didn't feel right.
The ceiling round box has 2 cables, both are 14-2 BX. One black wire is live-it goes dead when breaker is flipped off. Other black wire is dead. White wires are twisted and capped. The wires don't have any corrosion and insulation is in good shape.
At the switch, there are black and white wires. Both are dead with the breaker on.
The light fixture has 2 wires and a bare, braided copper (16 ga?). One was tied to the live wire and the other to the dead. Bare wire was on the ceiling box screw.
It looks like this fixture was wired so that the switch is on the neutral side. That is, by closing and opening the path on the neutral side, the bulb was turned on and off. This means the fixture was energized all the time.
Did I reason it right? Is it ok to use it with the fixture energized with the swtich in the off position? Feels like the new rules would ban this.
How to fix?
1. tie the live and dead black wires together in the ceiling box - take the power to the switch.
2. disconnect white wires in the ceiling box. One of them now becomes power feed wire from the switch (I'll mark it).
3. connect light fixture neutral to the neutral.
Is this good? I will check the continuity from the ceiling wire to the switch wire.
Thanks in advance.