Yes. But then I mostly ride mountain bikes and run drive trains until they're completely worn out. I used to do the whole measuring chain wear thing, and change chains regularly, but unless you changed the chain after a few hundred miles, it would just skip/suck, and you had to change everything anyway.
It works out more economic to run everything until its worn (current drive train on my main mtb is upto around 1200miles and is extremely worn but still functioning reasonably well) than risk changing the chain only, and have to replace everything long before then.
Off course, road bikes last far longer and usually still have more than one chainring, so depending on use, you might not have to replace both chainrings, but the same principles apply.
Regardless, if you look at the OP's photos, the small ring is obviously worn. I'd guess the new chain rollers are catching every tooth producing the clicking.
You might get away with just running in the bigger ring to wear the chain a bit, but certainly a new little ring would be my preferred option.