There has certainly been much debate about this topic in ultra running. Until late 2012, I sat pretty much on the side of the fence that said "runners should run." I used to think there were just two exceptions:
(a) When in need of non-impact training and
(b) To aid recovery (increasing blood flow for healing)
Thus, cycling at an easy pace may augment recovery after a long run and using an elliptical trainer or rowing machine may augment training without the impact on the legs that running requires.
In addition, I would have agreed that there are a few "hidden" benefits of cross training:
(1) having a sport to fall back on if injured and away from running or when engaged in a "rest" period (a.k.a., "off-peak fitness" period) and
(2) having a "fun" means of getting exercise and keeping fit when running might not being doing it for you (e.g., playing soccer, squash, climbing).
From late 2012, however, I started to encounter mounting evidence to suggest that regular lightweight upper body resistance training can improve running economy and oxygen efficiency. I also read that ultra running legends Ann Trason and Yiannis Kouros regularly weight trained their upper body with light weights.
To be clear, I'm not talking about body building, which purposefully adds a lot of weight. That would translate to more muscle to haul around a race course. However, the literature really paints a picture that strength work for the core, including hips, back, and abdominals, as well as the many small muscles and tendons in our bodies is probably essential for most, if not all, runners (See Biomechanics page for more on strength work).
From 2013 through 2015, more research appeared on "maximal and explosive/plyometric" strength training benefits. Thus, from 2013, I began my own personal experiment with this type of strength training a few times per week, whilst watching the literature. My own subjective measures of performance changes, based on my race results since 2013 and accompanying injury-free status have me pretty convinced! One of the key benefits I have seen is that I maintain an upright stance through 24 hours or more of running; a hunched-over body means a compressed diaghragm not taking in adequate oxygen.
Strength training is no longer "cross" training to me, but is an essential component of ultra distance running training.
Cycling? Skiing? Ellipticals? Well, that probably goes back to the first points. Runners run. And lift weights ;)