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Shouldn't it be the continuous form 'is drawing'?

Depends on context

In which way?

As the year draws to a close, the family is ....
The year is drawing to a close.
Can't give you a grammatical reason - just sounds better to my ear.

The reason why I started to wonder about simple present tense or present progressive is the hint: 'weihnachtlicher Gruß'. I am never sure myself, but what I learned is: We use the present progressive to talk about temporary actions and situations that are going on 'around now', e. g. "Why are you crying? Is something wrong?"***
We also use the present progressive to talk about developing and changing situations, even if these are very long lasting, e. g. "That child's getting bigger every day". "The climate is getting warmer." ***
Does this correspond with our 'unsolved query'?

Grammatically you are unquestionably correct. However, depending on the subtlety of meaning, where you want to put the emphasis etc. etc., it can at times sound stilted and is not the only solution.

++ In the above example, it is good. With as, I would not use it.

I agree with Claudia in both examples she gives, and I would not leave out "as" in the first one. (native US)