What are the advantages of wave skipping?

Only spend more stars to skip waves if you want to set a record. Not sure if skipping waves gives you shards

also you play less games if you skip waves. I recommend that you pay 5 stars as stars can be earned every 2 days. Skipping waves makes you get rewards faster but you can not play as much as just using 5 to start at wave 1. 

Personally I think it's there to allow better players to help people get the trophy achievements.

Someone may not want to go through all 25 waves to help someone who is really bad but wouldn't mind going through a couple...

It quicker to get the 25k shard minimum for the best chest if you start at a higher wave.

You're better off starting at wave 1. The only advantage you kinda get from starting at the higher waves are more shards from enemies (the amount you get increases in the higher waves) but considering that the higher waves are harder and you need to at least break 25,000 to get the highest possible chest you're better off saving the extra stars and going through it on the easier waves because i don't think you save that much extra time by starting up in the higher waves but i could be wrong on that since i never tried it.