This is a very complicated and delicate matter. Germans as people are not only the people living in Germany but also those living in Switzerland, Austria, Netherlands, Belgium, Letzenburg and even large regions of France and other countries. Although the Germans in the east were driven out of the eastern block by the soviets, but this is not relevant for the western parts.
In history this is most notable and easy to verify how languages are spread through times/regions/dialects/etc. A good map which people are all 100% officially German:
https://de.wikipedia.org/wiki/Westgermanische_Sprachen#/media/File:Westgermanische_Sprachvariet%C3%A4ten_1990.png
As some of the previous comments pointed out, the "van" means "from" and "von" is usually related to royal or higher social rank. However, the "van" is not only dutch. There are also many families in Germany using the "van" in their names. As "van" is 100% German origin just like "von" just a different meaning and not widely spread in Germany as the "von"
Actually there are no real "dutch" words. They are all "mittel hochdeutsch" or descending. Also the dutch grammar was until 1905 simply put German grammar. Many people also think: German = Wandern and Dutch = Wandeln. Actually both is simply put German. In Germany however the "modernes Hochdeutsch" evolved different and with a lot from southern Germany where "dutch" being spoken today in the Netherlands and Belgium evolved from "mittel Hochdeutsch" without this influence from the south and with many words from northern Germany. Even words like "ik" which is "I" in English and "Ich" in modern Hochdeutsch are nothing dutch but simply German and being spoken in German dialects as far as close to Berlin.
Just some example to point out, Dutch and Belgium people today are different nationalities but not different people or origin! THey are all "Germans". So if you speak of "Van Beethoven" as "German" then yes, 100% German race and German origin.
Inside the Germans of course, you have many dialects and many tribes: Frysien, Sachsen, Bavarians, Zeeuwen, Vlamingen, Holländer, Rheinländer, Limburger, etc. Some of those are now together and Form a country called "Germany" and some are together and form a country called Belgium and some are together and form a country called Switzerland and some are together and call their country the Netherlands, etc. Heritage-wise all Germans though.
Some are even devided. So the people from Limburg, with their dialect, like me, are separated over 3 countries:" Germany, Netherlands, Belgium" I did a lot of Genealogical research and found my ancestors both come from Limburg which today belongs to Belgium, the Netherlands and Germany! So it used to be 1 people, 1 dialect, but separated by royals over time! Many wars been fought over Limburg. Where in fact, historically and from inheritance it's all 1 people divided by royals, kings, counts, etc.
So, the only thing, if you don't want to call Beethoven a "German" that remains are really the modern nations and nationalities. But did they exist at the time Beethoven was born? The answer is no. There were no nations called "Germany, Belgium or Netherlands" There was a Holy Roman Empire which throughout history included mostly the modern Netherlands and Belgium. At the time of beethovens birth, this part of todays Belgium was simply a part of the holy roman empire. But there was no Germany nor Belgium at that time.
Also the United Kingdom of the Netherlands (not to falsely mistake as the same as the Kingdom of the Netherlands, as they are not the same) was most part of history part of the holy roman empire. A unification of all German tribes in Europe. So at the time Beethoven was born, there was no Belgium or Germany or Kingdom of the Netherlands.
Prior there was a Republic of the United Netherlands, which Belgium wasn't a part of mostly, but at some point Belgium and the Netherlands were together. But then again, it's very complicated. As for example Limburg (today divided into Belgium/Germany/Netherlands) was officially part of the United Kingdom of the Netherlands and later Kingdom of the Netherlands until 1867 and at the same time it was part of "Germany" which didn't exist as a country, but it was part of the Frankfurter Parliament.
It's really complex, the history. Anyway, at 1770, there was no Belgium nationality, nor German nor Netherlands nationality. There was no modern Netherlands/Belgium/Germany. According to modern standards, you couldn't say: a person from that time was from Germany, the Netherlands or Belgium. All you can say, a person from that time is German origin.
That's why historians consider Beethoven "German" but not as if he's from Germany, but he's German in the same way all people from Belgium, Netherlands, Letzenburg, Austria, Germany, Switzerland, Alsace, and few other areas are "German"