I'm afraid that the absence of the regular theta is not a bug. It is, rather, a limitation of the font itself which simply does not include it.
The mathematics support is based on the italic, as the documentation explains. I can't find any evidence that the regular theta glyph is provided by the italic. The specimen linked in the question does not appear to include it:
The regular theta you see in the specimen is shown only in the upright shape:
ebgaramond-maths does not support upright Greek. Not only is italic Greek standard in maths, as I understand it. Supporting upright would require a significant amount of work, which would really not be worth it for a package which inevitably provides at most partial support for mathematics.
The only thing you could do is make a feature request to the developer of the font, asking that the regular theta be added to the italic. Given that the font is based on a historical one, I don't know whether the developer would be sympathetic to this request or not. (For example, I don't know whether the lack of the regular theta from italic reflects the standard at that time or not in the way that the lack of bold reflects the absence of bold type during that period.)