So it looks like the certificate had something to do with it; I exported it from the server and installed it on a workstation and am now seeing more task logs, but it still doesn't seem to be registering to the Task Server. Here are some of the logs I'm seeing:
<event date="09/23/2015 11:50:56.4680000 -04:00" severity="8" hostName="COMP7" source="Client Task Agent" module="client task agent.dll" process="AeXNSAgent.exe" pid="6860" thread="7136" tickCount="681724649"><![CDATA[CTaskAgentBase::ProcessTasksThreadProc(): Registration is in progress, check for tasks is postponed.]]></event>
It flipped between these two states a couple times:
<event date="09/23/2015 11:50:56.4710000 -04:00" severity="8" hostName="COMP7" source="Client Task Agent" module="client task agent.dll" process="AeXNSAgent.exe" pid="6860" thread="7136" tickCount="681724665"><![CDATA[CTaskAgentBase::_SetAgentState(): Change of state [CTA_STATE_CheckingForTasks] to state [CTA_STATE_NotRegistered] is allowed.]]></event>
Then these failures as far as registering:
<event date="09/23/2015 11:50:56.4710000 -04:00" severity="8" hostName="COMP7" source="Client Task Agent" module="client task agent.dll" process="AeXNSAgent.exe" pid="6860" thread="7136" tickCount="681724665"><![CDATA[CTaskAgentBase::_IsRegistered(): result [not registered].]]></event>
<event date="09/23/2015 12:36:32.6180000 -04:00" severity="2" hostName="COMP7" source="Client Task Agent" module="client task agent.dll" process="AeXNSAgent.exe" pid="6860" thread="9712" tickCount="684460689"><![CDATA[Failed to call web interface by url [http://AltirisServer.Domain.com:80/Altiris/ClientTaskServer/Register.aspx?lastResort=true&resourceGuid=0b29e1b5-db66-4d1f-a7e0-6c3df676860c&crc=00070006001A064F], error [0x80042D21, IDispatch error #11041].]]></event>
<event date="09/23/2015 12:36:32.6200000 -04:00" severity="8" hostName="COMP7" source="Client Task Agent" module="client task agent.dll" process="AeXNSAgent.exe" pid="6860" thread="9712" tickCount="684460689"><![CDATA[RegisterWithTaskServerOp::_register(): Completed with [FAILED (may retry)].]]></event>