Polonium 210 was used in the very first internal neutron initiators - Urchins and the like. Back then concern about storage of rockets for decades was not a priority. ICBM also did not existed (first ones were in 1957-58, 12 years after first nuclear test), nuclear bombs were supposed to be carried by long-range bombers - and hence could have been serviced before flight. Finally, war could have started next week - and everybody rushed to any solution which could be made faster.
Newer designs used external neutron initiators based on tritium/deuterium linear accelerator and deuterated/tritiated lithium target. Tritium-based ones are much easier to start, but also have limited shelf life. Deuterium-based initiators have no nuclear-related longevity limits.
You can have a hint on how neutron initiators operate by looking at specs of Dynamic Albedo of Neutrons device of Curiosity mars rover.