I think not, in every case not if you have a dragon summoner as you can play with him also your 2 splinters. But in the end you need only the Dragon Summoner and no Dragons, as you can play dragon summoner without dragons.

Let's calculate:

Now you have 2 splinter from 5, so your chance to get the "right" quest is 40%.
In the 60% cases you get other splinter, you change the quest and (as it remain only 4 splinter and from this are 2 "your" splinter) have a chance of 50% to get now the right quest. So 50% from 60% means 30%.
Your chances to get (with one change) the "right" quest is 70%.

Later you have 3 from 6 Splinter, so the chance to get the "right" splinter are 50%.
In the case you don't get the right quest you change and your chance is now 60% (3 from 5 available splinter are "yours") - so it means in 30% of cases you get now the "right" splinter.

So your chance is later 80% instead of 70% to get (with one change, if neccesary) one of your splinters.