This problem is the result of WiX installer not properly installing the custom tasks. Below is what I did to verify this.
For test purposes I created an empty custom task and manually deployed it to the target server by copying the files and the DTS\Tasks directory. This worked so I added the .cs file of one of the tasks that was throwing errors, to the test custom task and re-deployed. I was able to see both tasks. After cleaning the solution with the original custom tasks, I manually deployed the original custom task to the target server, like the test custom tasks and was able see the tasks in the SSIS toolbox. I then uninstalled the custom tasks and deployed them to the target server using a WiX installer -- the problem resurfaced. So the problem is with the installer. My previous attempts at manually installing the custom tasks failed because I was using the assemblies installed from the WiX installer, although I don't know why.