I'm not sure I go along with that fully.  I wouldn't imagine sidechains having a very easy time of it with native Bitcoin in a state of turmoil in my conception of things.

Where I really see sidechains helping would be before troubles started.

For one, sidechains have a powerful incentive to keep Bitcoin working at least well enough for themselves to operate, but probably also well enough for everyone in order that Bitcoin remains credible as a backing store.  Indeed, as I've already (kinda) proven, straight forward simplistic mining does not have the long term potential to be profitable due to supply/demand issues with mining gear.  Sidechains would be almost forced to provide support for Bitcoin no matter what the profitability of doing so might be.  So this might provide a credible support foundation (as opposed to miners tapping other revenue streams like milking the userbase for intelligence data.)

For two, I could see sidechains acting as a force to split up mining unions as they might prefer merge mining different sets of sidechains and what-not.  I honestly don't know enough about it to know if that's a realistic hope though.