Is there any benefit to having multiple recruitment radio beacons?

No, the game only checks if there is a beacon in a settlement.

Only happiness increases the chance.

Build bars ( lvl 3 restaurant) to get +40 happiness.

No new settlers will come if you have five settlers that are not assigned to anything.

Personally I didn't notice any difference when I built multiple towers, but I haven't properly tested it

Make sure you have excess food, water and defence (you need 1 food per person, 1 water per person, and 2 defence per person (so as to keep your defence rating equal to the sum of your food and water)) and people should start arriving fairly quickly

Also note that there's a population cap on each settlement, which is equal to 10 plus the number of points you have in the charisma SPECIAL attribute