You are asking the wrong question (or at least, a wrong question). It's not "why does the US need any involvement", it's "Why has the US insisted on involvement for so long?" (e.g. during the 1980s when widespread sentiment in Europe was for the US to close its military bases, the US insisted on remaining).
The UK does not rely 136k people to defend itself from military risk. It relies on its nuclear arsenal, which while not as large as those of the USA or Russia, it quite the deterrent all by itself.