You have to hear it in the context of what else was going on at that time and what had been going on in the preceding years. And, with the benefit of hindsight, what it started.
While "Love Me Do" was a hit, it was not, as "spirit ..." showed, a big hit. It may have been released in October 1962 but it wasn't until December that it reached #17 in the RECORD RETAILER chart, by which time The Beatles had almost a month out of the national public eye, with a 13-night residency in Hamburg and a short tour of northern Scotland. It was really only after those that The Beatles released "Please Please Me" and began to build their national reputation - more prestigious and larger venues and more radio and TV shows, for example. By the spring of 1963, the very notion of popular music in Britain had begun to change (beat clubs opening all over the place and the charts no longer being filled with American hits) and, by the middle of 1963, the national press began to feature The Beatles, and beat groups generally, in a more consistent and more synpathetic way, while, by then, the weekly music press and the teenage weeklies (mainly for girls, it has to be said) were providing blanket coverage.
"Love Me Do" may have been a simple, almost pre-teen, song but it really did presage huge changes in Britain.