Based on @Donald Byrd's answer, I have created a more optimized version. There's no need to recreate the entire structure—just ignore default or empty values.

public class CompactJsonExtantFormatter : CompactJsonFormatter
{
    /// <inheritdoc />
    public CompactJsonExtantFormatter(JsonValueFormatter? valueFormatter = null) :
        base(valueFormatter ?? new JsonExtantValueFormatter(typeTagName: "$type"))
    { }
}
/// <inheritdoc />
public class JsonExtantValueFormatter : JsonValueFormatter
{
    private readonly string? _typeTagName;

    /// <inheritdoc />
    public JsonExtantValueFormatter(string typeTagName) :
        base(typeTagName)
    {
        _typeTagName = typeTagName;
    }
    
    /// <inheritdoc />
    protected override bool VisitStructureValue(TextWriter state, StructureValue structure)
    {
        state.Write('{');
        char? delim = null;
        foreach (var prop in structure.Properties)
        {
            if (IsDefaultValue(prop.Value))
                continue;
            if (delim != null)
                state.Write(delim.Value);
            delim = ',';
            WriteQuotedJsonString(prop.Name, state);
            state.Write(':');
            Visit(state, prop.Value);
        }
        if (_typeTagName != null && structure.TypeTag != null)
        {
            if (delim != null)
                state.Write(delim.Value);
            WriteQuotedJsonString(_typeTagName, state);
            state.Write(':');
            WriteQuotedJsonString(structure.TypeTag, state);
        }
        state.Write('}');
        return false;
    }

    private static bool IsDefaultValue(LogEventPropertyValue value)
    {
        return value switch
        {
            ScalarValue { Value: null } => true,
            ScalarValue { Value: string s } when string.IsNullOrEmpty(s) => true,
            ScalarValue { Value: 0 } => true,
            ScalarValue { Value: 0L } => true,
            ScalarValue { Value: 0.0 } => true,
            ScalarValue { Value: 0.0f } => true,
            ScalarValue { Value: 0m } => true,
            ScalarValue { Value: false } => true,
            SequenceValue seq => seq.Elements.Count == 0,
            DictionaryValue seq => seq.Elements.Count == 0,
            StructureValue structVal => structVal.Properties.Count == 0,
            _ => false
        };
    }
}