That so-called "sloppiness" was part and parcel of what Zeppelin was best known for: improvising. Their live shows were legendary. Zeppelin and U2 are similar in that they were often better live than in the studio. If you want to hear Page at his best, get their # 1 live album "How the West Was Won"...it's pure magic, and I have never heard any guitarist play as spectacularly as Page - even Hendrix - then in these 1972 shows. (BTW, don't use Song Remains the Same as a yardstick...Plant's voice was beginning to decline by then)
As well, no one could construct a catchy memrable riff like Jimmy Page. Even the Edge in a Rolling Stone Interview about Zep in the late 80s acknowledged he practised Page riffs during rehearsals. In addition to many hard rock/metal guitarists, Page's guitar style influenced everyone from Pearl Jam, the Pumpkins, and STP, to the Stone Roses, Jane's Addiction, the Beastie Boys and numerous others.
Cheers
PS. Make sure you pick up Physical Grafiti...Stairway may be their most famous song, but Kasmir was the quintessential Zep song.