QT 6.5.2 is really best on 10.3.x and FCP HD. QT 7.02 -03 is best on Tiger with FCP 5.
The issue with FCP failing to startup can be as simple as running Disk Utilities> Repair Disk Permissions or perhaps more troubleshooting may be necessary.
What was done to these computers last before they started exhibiting the problem.
Generally, you should run Repair Disk Permissions after installing software,