When I squeeze the brake levers fully it doesn't touch the handlebar, so that's not the bottleneck.
How can I improve the braking power of such a drum brake?
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As bicycle mechanic in the Netherlands, I see a lot of these brakes. The drum shoes can become harder due to wear or age or something. They can also become a bit uneven.
If the drum is to worn out, you can easily replace it.
This is just like an automobile brake and would be fixed using the same techniques.
First make sure the lever is moving freely, and is not "bottoming out" due to the cable being too tight. (To adjust the brake you do not tighten the cable but rather adjust the eccentric that is opposite the lever pivot. Then adjust the cable to match.)
Lubricate the pivot lightly, but do not get any lubricant on the drum or shoes. If need be, wash off any oil or grease on the drum/shoes and wipe with a naptha-moistened cloth, then sand the shoes lightly to remove any glaze.
Examine the shoes to make sure they are not worn out, and examine the drum for serious scoring.