As an archaeologist, I can tell you that wooden bowls and spoons did exist and were used. The problem we have when researching them is, that they rarely survive till today, because they need very special conditions to do so.

Either wooden objects have to be waterlogged without oxygen, or they need very dry conditions with stable temperatures, or they will disintegrate. When they are found, both of them, the wet and the dry finds need special care and treatment to prevent disintegration and allow a presentation in a museum. Horn objects are more durable, but also these won't necessarily survive till today. Furthermore, to notice such things as traces of use and wear or varnish, you would need well preserved items and study them under a microscope.

That stated as preliminary, we do have finds of wooden plates, bowls and spoons from different areas and timelines. To give a few examples (not a complete list by far), in the National Museum of Archaeology in Dublin there is a permanent exhibition for the Viking times, and they lifted a beautuful amount of everyday wooden items out of the River Liffey, where they survived in the mud.

Another Viking time example are some wells or latrine pits in Viking Age York.

For Germany, there is a socalled "wurt" called Feddersen Wierde where wooden objects were found. There was an exhibition last year in Bonn which presented some half-turned bowls, so we even can make assumptions about their manufacturing process. Feddersen Wierde is a settlement in use between the First Century BC and the Fifth Century AD and is a Old Saxon settlement.

In the South of Germany, there are two cemeteries with an Early Medieval context from the Alamanni, which have an extraordinary well preservation of wooden objects, including furniture and a lyre. The places are Trossingen and Oberflacht.

So, archaeologically studiable items mostly come as grave goods, or in rubbish layers and/or pit fillings. While the first mostly are whole and could still be used, the latter were likely thrown away or discarded. In case of deep wells an accidental loss is also possible. This makes a detailed study of wear and use more difficult. But, you can study which kind of wood was used and if fresh wood was used or if the piece was stored and dried before its use. This mainly is done for building parts, because there are bigger pieces of wood for analysation.

And I don't know if you use wooden spoons for your cooking today, but they really don't need varnish. They will be smoother if you give them a pinch of oil some time or other, but you can use a simple plant oil for this, and that was known and used. If you wish to polish wood, a simple mixture of bees wax and some oil to make it more applicable works just fine, but I am not aware of any archaeological find where this could be traced. However, wooden items can be cleaned just fine. Horn objects should not come to touch very hot liquids, but otherwise, they can be cleaned with moderate warm water as well. I am not aware of any study focussing on the duration of use for wooden or horn objects in my field of study though. If there are any, you might want to look into the complex of "experimental archaeology".