Replacing/Upgrading brakes on ‘80s Peugeot road bike
You can mount recessed style brakes on your 80's frame by drilling out the hole through which the brake bolt goes on the backside of your forks such that the recessed style bolt fits through the hole. 

You may have to get long reach style brake calipers in order for the pads to be able to reach the rim. These would be an option: https://www.amazon.com/Tektro-R559-Calipers-Silver-55-73mm/dp/B006Z0OVWC
You can find cheaper models but in general the newer style (dual pivot) long reach brake calipers are more expensive than the standard reach models. You could potentially look for a second hand pair.

Another option (If you want significantly more braking power) is to replace the fork with a newer model which will work with normal range dual pivot calipers. These calipers have better leverage and therefore better braking power.