It might seem like that flat C sharp doesn't matter since it's over quickly

First of all an apology. I had forgotten the tune was in D and I was thinking Cnat to E rather than C# to E and the suggestion I made was for an alternative Cnat fingering not a C# fingering.

As to alternative fingerings, they can be very useful so I would not dismiss them out of hand. I agree it is important to be aware that some alternative fingerings are more in tune than others and there is a need to be aware accordingly. The alternatives I gave for E are all in tune and are extremely useful and I use them all the time.

The suggestion I gave for D - E was taught to me by a classically trained professional recorder player as a D/E trill and it is both much easier and more in tune than the one given in most fingering charts. It is also useful for playing a roll on D as it only involves moving one finger and the thumb.

I agree with Anahata's comment and would add one other thing - listen carefully, especially when playing with others. The recorder is a very sensitive instrument and it is possible to make small adjustments in intonation as you go