I don't think one should start with the presupposition that the beginner to LvB quartets is deeply familiar with Mozart and Haydn. That's why I doubt that op.18 is usually the best option. Most relative beginners start with romantic and classical symphonic music like the best known symphonies by Beethoven, late Mozart and Haydn, Dvorak, Tchaikovsky etc. They are used to big, bold, highly emotional stuff, so for many the mere sound of a string quartet will often be something they have to get used to. That may be one reason why Schubert's d minor "Death and Maiden" is among the most popular.
So usally a middle period Beethoven piece would be best (and there is probably no doubt that among his orchestral music and maybe also piano sonatas the middle period pieces are the most popular) but as I said above, I have my doubts. Maybe op.59/3 is actually the most popular quartet (it seems to be very often on anthology discs and the beginning of the finale was used many years in a German TV show on literature) but I do not find it as emotionally appealing as e.g. 59/2. And this one has a rather harsh first movement.
Overall, I am not at all sure that the middle period quartets are easier than the late ones. And the late ones are often emotionally very immedate. Sure, the very form of op.131 could be confusing but that's less a problem for the beginner than it was for Beethoven's contemporaries.
Anyway, the corpus is not that intimidating (compared to Bach cantatas or Haydn symphonies) and nowadays the whole bunch can be listened to free on youtube or bought for $25 or so in a cheap box, so the easiest way for a beginner is simply to try a few works and keep going and it does not matter all that much where one starts.