In my daughter’s case, demonstrating interest just wasn’t that big an issue because she was truly interested in each and every school on her list. On a superficial level, I did take her to college fairs and meet-and-greets so she could sign in and she requested material via the website. If D couldn’t think of any questions after viewing a school’s website or talking to a college rep at a fair, that was a sign that she wasn’t all that into the school. Inevitably, she dropped that particular school from her list.
If nothing else, questions arose almost immediately when she looked at the CA. She would email questions about their questions If she couldn’t find an answer on their website. In one case, the admissions person called back. That took her by surprise - she wasn’t expecting a phone call, didn’t know the area code so she didn’t pick up. Hearing the message, she called back, apologized and then the two chatted a bit about the school. It may have helped that she only applied to small LACs that were very responsive and personable. She didn’t visit every single school before applying but she talked to every single school before or during the application process.
I have no idea how she would have dealt with large research universities - then again, I think that’s exactly why she didn’t apply to large research universities.