This bike's brake performance is very poor due to design and materials being used.
It has already been suggested a brand that sells replacement pads. New, soft rubber pads should make it a bit better.
My experience with rim brakes tells me that keeping the brake contact surface on the rim clean can also achieve modest performance increase, but on steel rims it's even more limited. I usually wash rim surfaces with dish soap.
After that and a proper tuning (tightening the rods to the correct length) you'll likely get the best possible out of those brakes. (which may not be enough).
I once assembled a dutch bike (Gazelle) with similar shape and design to the Flying Pigeon. I used cheap caliper brakes on regular clincher rims for 700c x 38 tires. (I received only the frame, fork and handlebar with only part of the rod system). The bike had no mounting points for the calipers, so I fabricated them out of angle iron and positioned in such way that the braking force would press the angle iron against the seat stays or the fork legs.
It was a fiddly work and took some time to tune properly but got decent performance out of cheap caliper brakes actuated by rod levers. That could have been a good prototype if I had been willing to weld the mounting points to the frame and make it permanent (The frame was very good quality steel). Permanent mounting points would have probably eliminated some flex and made tuning easier and the whole system more performant.
However, that endeavor was only feasible because I had to buy wheels anyway, so I choose rims with braking surface in a size that would allow a wide range of compatible tires, had enough spare time and experience fiddling with and modifying brakes. I also really wanted to ride that bike. (700C x 38 tires are virtually the same size of 28", Maybe 4 mm 1/4 inch smaller diameter, but the rims have a different shape. 28 and 700C tires are not interchangeable and one cannot fit a rim sized for the other.)
If I had to replace wheels and hire a frame builder to add the brake mounting points, and a mechanic to install a whole brake system, the cost would just had been enough to buy a complete bike. Maybe a used steel road bike would have been a near good replacement.
The only gain I perceived out of my experiment was the comfort of that ride. The resulting bike was an excellent commuter, soft ride, not tiring on longish rides considering its weight and a fairly capable bike for carrying loads. The brakes where good enough to control the bike down a slope while ladden with groceries.
In conclusion: If you do not have special reasons to retain that bike you may be better off selling it and buying a more modern design, if budget allows. If not, then try to replace the pads, clean the rims and adjust the rods as best as possible. Anything else may not be cost effective.
Regarding the lights the best option really is to use a battery or rechargeable L.E.D lamp. There are many options ranging from dirt cheap to expensive and powerfull lights. A cheap LED on fresh batteries is many times brighter than a 6 or 12 volt incandescent bulb and have flashing modes that are better for alerting other road users of your presence.
There may be retro fit LED bulbs that are compatible with your dynamo, if you wanted to keep the look and feel.