Full disclosure: I did the National Trust tour of the Beatles' childhood homes, but did not do the Magical Mystery Tour.
However, I think if you're not Beatles fans, you may prefer the childhood homes to the MMT. The childhood homes tour, as said above, is very much about the way people lived in that time and place (postwar suburban Liverpool), and there is a lot of material about the contrast in the differing financial and parental circumstances of John and Paul. It goes way beyond music; music isn't even the main focus. In fact, in my group, I can't recall any of the participants mentioning songs at all. In other words, we weren't superfans, asking "was Please Please Me written in the kitchen or the bedroom?" or the like. The docents mentioned such things in passing, but gave much more detail about the houses, families, etc, and that's what we were asking questions about.
Again, I haven't taken the MMT, but from what I understand there's a heavy focus on locations mentioned in song lyrics and the like. It may be like someone who's never seen The Sopranos taking a Sopranos tour in New York (yes, they have these), or me, who's never read the Harry Potter books, taking a Harry Potter tour.
If you just want an overview of Liverpool by bus, take a Hop On Hop Off bus (HOHO). There are two; at least in 2016 when I did this, the City Explorer had live guides and was £1 cheaper than the Liverpool City Tours, which had recorded commentary.
Should you wish to avoid Beatles sights altogether, you could easily fill two days with some of the following:
The Museum of Liverpool (although I found this a bit of a jumble, it is free)
The Merseyside Maritime Museum and International Slavery Museum (free, very worthwhile - I saw it over two visits)
Tate Gallery (parts are free, parts will have admission)
Liverpool Cathedral (Anglican) and Metropolitan Cathedral (Catholic) - these two are a nice walk apart on Hope Street, and couldn't be more different.
World Museum
Walker Art Gallery
I had four nights in Liverpool, and didn't even get to the last two (I did the Beatles Story, which took time).
I also agree with a day trip to Chester. This is easy to do on MerseyRail, with trains every 15 minutes Monday through Saturday and every 30 minutes on Sundays. These local trains leave from four downtown stations (Moorfields, Lime Street, Central, and James Street), so it's easy to find one near your hotel.