Would the nuance of the context changed if I remove "so" here?
Yes, it does change slightly! "So" is used for emphasis here, and to make the sentence a little more friendly and casual. If we were to remove "so," "I'm glad you're here" is a little more reserved in feeling. English speakers typically use "so" often when speaking, even if there isn't really a need for it. Ex: That outfit looks so good on you! Ex: It's so hot outside.
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