The modern chain that's on there is much narrower than the one that left those marks on the side of the cogs.

This chain will be more forgiving of shifting error, because it's narrower and because it's also much more flexible. The inside width is plenty wide also.

But those backwards-pointing sharpened corners at the tips of the teeth of the third-largest cog need to be beveled slightly for a new chain to transmit much chain tension without skipping.

Picture the chain feeding onto that sprocket, with a roller between two teeth, and with the next link and roller pivoting to fall between the next two teeth. Notice how the pointed corner is going to interfere with the roller, then grind a bit of that metal off.
Below is a photo of a generous bevel on the driven side of one tooth stamped "ba". A smaller bevel, angled just a bit more "vertically" along the driven side of the tooth, is what you should be trying for.
The teeth to the left and right of the "ba" tooth (stamped "12t" and "bd") are still very sharp almost like yours.
Your vintage cog teeth are taller than on the modern one pictured, which shows the wear in a somewhat different proportion, but since you have skipping under load, it has certainly reached it's wear limit and needs correction in order to transmit chain tension without skipping.