It didn't work out
I completely forgot to update on it. It didn't work out. The bank refused our loan proposal. It was a tough period for us and I guess for economy in general. I think they have tightened their budget and ran on conservative policies. The broker company I was working with was surprised to see some of their customers including us rejected. According to them, many of them are from non-EU countries. After the contract got canceled, we gave it up.
And this February we tried to start the process again. However after the broker talked to a bunch of banks, it became clear. A banker told them
Your client's visa will expire in less than 2 years. Tell them to come back when their visa gets renewed.
We hated the situation so much, but we couldn't just stop there. So we moved to a new place even further from Paris. It's still an apartment but with a great garden and view. We are having an amazing time here. Our lifestyle has already changed a lot in positive ways.
In some years, we will start looking for a house again. Let's see how it goes!