... I was able to listen to just the Commanders side of the conversation because I was only working with a Yaesu FT-60R...
The reason you couldn't hear the uplink is not because of the radio you are using but because you are not within line of site of the transmitting station (school) and the uplink frequency is kept confidential to keep the ISS from getting slammed while crew members are talking to a school.
Your radio is very capable of communicating with the ISS. It is just a matter of catching a crewmember on the radio when they are favorably placed for you. This doesn't happen very often as they don't have a lot of free time and many of them are not very active.
ARISS contacts are very exciting for school students. The process and requirements to get one scheduled are demanding and expensive but well worth it. Why not look into it more fully and try to help a local school set one up.
Fun, fun! Keep listening.