でしょうに is でしょう for inference, followed by the conjunctive particle に meaning "but". This に is the same に used in [のに][1]. You can think of でしょうに as a variant of のに but with an added nuance of inference. Its plain (non-polite) version is だろうに.

- 明日は雨が降るのに、出かけるの？ (informal)  
  Although (I know) it rains tomorrow, are you going out?  
- 明日は雨が降るだろうに、出かけるのか？ (blunt)  
  Although it's likely to rain tomorrow, are you going out?
- 明日は雨が降るでしょうに、出かけるのですか？ (polite)  
  Although it's likely to rain tomorrow, are you going out?

> まだ そんな心配する年じゃないでしょうに。  
Although I think you're not old enough to worry about that(, why do you say that)? / But I think you're not old enough to worry about that!  

There is no meaning of "obviously/当たり前" in your sentence. In the linked question, such a nuance comes mainly from そりゃあ (=それは). Its literal meaning is close to "As for that" or "When it comes to that", but by extension, it can express the nuance of "it's only natural". For example:

- 「明日はサッカーを見るの？」「それは見ますよ！」  
  "Are you watching soccer tomorrow?" "(When it comes to soccer,) Why not!"
- https://japanese.stackexchange.com/q/82516/5010

  [1]: https://jlptsensei.com/learn-japanese-grammar/%E3%81%AE%E3%81%AB-noni-meaning-even-though/