However, recently it's been skipping when I pedal — suddenly the pedal will just jerk down and the chain seems to slip past a few teeth. I've concluded that the crankset probably needs to be replaced.
This conclusions is most likely false. A chain does not slip on front chainrings.
There are two reasons why the chain could seemingly slip:
First, a chain can slip on rear cassette sprockets. This most often occurs when installing a new chain after the old chain has worn cassette teeth in a manner that cause them to worn only on the old chain. However, I suspect if not changing the chain for a long time and letting it wear way past the recommended wear limit of 0.5%, it might be possible that the old chain could start slipping. I have not experienced old chains slipping because I always replace my chains at 0.5% wear, and occasionally notice the chain slips so the cassette sprockets need to be replaced too.
Secondly, the freewheel in the freehub body could slip. This most often occurs in cold conditions on freewheels that have been lubricated with thick grease. The cold causes the grease to become even more thick, so thick that it prevents the pawls from engaging the freewheel ring. The fix is to remove the grease from the freehub body and install a light oil in its place. Unfortunately, freehub bodies can be difficult to open.
Since you said you replaced the chain but did not say about replacing the cassette, I suspect your slipping is of the first type, and the fix is to replace the rear cassette sprocket set.
The rules of drivetrains are:
- A new chain may not engage to a worn cassette
- A worn chain may not engage to a new chainring
So when replacing any drivetrain component, you may in some cases need to replace everything behind it too. Example: if you replace a chainring, it might necessitate replacing the chain, which can in turn necessitate replacing the cassette.
I would never replace crankset or chainrings because of a slipping chain. The best chainrings (7075T6 / 7075T651 aluminum alloy) can last hundreds of thousands of kilometers. The only reason to replace a chainring is if the teeth have been practically worn away, or if you experience chainsuck, i.e. chain not disengaging from the chainring (happens only with long cage rear derailleurs and small chainrings, typical of MTBs -- does not happen on road bikes).