I am a native speaker and I have been on both sides of the table. In my opinion, “if we were to offer you the job” sounds perfectly natural, and there’s no need for you to cringe.
In some cases, multiple people at the hiring company will interview the prospective employee. In the U.S. at least, it’s common for HR to brief these people to not give the candidate the impression that they can commit the company to making the hire. The actual decision to extend an offer will be made after all the interviews are complete and the results collated. In general, the aim is to avoid giving someone the impression that they would have been hired, and their failure to get the job was due to racism, sexism, ageism, ableism, or being in one of the “protected categories” as they are referred to in U.S. law.
There are many questions that could give a prospective employee the impression that they are close to being hired, such as “when can you start?”
It’s difficult to assess the language used by a specific interviewer, since interviewers vary in their understanding of employment law and how to best communicate with potential employees. It can be difficult for a candidate to gauge what the interviewer is “really thinking”. What looks like indifference could actually be interest that the interviewer is trying to conceal.
My first recommendation is that you read up on body language, which is often more telling than the words alone. However, in my experience, an interviewer saying “if we were to offer you the job” is a sign that they are entertaining the possibility, and gives you an opportunity to respond with warmth and enthusiasm, at least if it’s a job you want.