The downsides ive noticed are:
1) Price changes during mayhems on weapons, alot more people decide to sell due to demand.
2) Some resources can become limited during this time so you need to stockpile (either the resource or the item) to avoid spending more to make less.
3) Funds get tied up in general (not just during mayhems) as you need to make sure you do an okay run size to average out your costs. Which means you usually have more product than you wanted and have to wait for it to sell. For new crafters or those without a decent bankroll, this can stop their crafting for a while.
4) going along with #3, Bankroll is very important!
5) On VU updates, it can change resource availability and something you used to craft just suddenly became a lot more expensive to craft and may no longer be worth it. If you own a shop or focus on a certain type of item, this could mean you need to change your whole setup.
I don't find those that undercut for short periods of time a big issue, they usually can't keep up with demand anyway and your product will sell at usual price. It is something to keep in mind when listing your item(s) though. I have also undercut just to get rid of something that is wasting funds, usually those are items that i crafted for a reason (like skilling or a special order) and don't need the rest of the product that was made. Sometimes having funds sitting there doing nothing is worse than selling for less or even for a loss.