Why would Oracle.ManagedDataAccess not work when Oracle.DataAccess does?

Try to add the path to tnsnames.ora to the config file:

<?xml version="1.0" encoding="utf-8" ?>
<configuration>
  <oracle.manageddataaccess.client>
    <version number="4.112.3.60">
      <settings>
        <setting name="TNS_ADMIN" value="C:\oracle\product\10.2.0\client_1\NETWORK\ADMIN\" />
      </settings>
    </version>
  </oracle.manageddataaccess.client>
</configuration>

The order of precedence for resolving TNS names in ODP.NET, Managed Driver is this (see here):

1. data source alias in the 'dataSources' section under section in the .NET config file.
2. data source alias in the tnsnames.ora file at the location specified by 'TNS_ADMIN' in the .NET config file.
3. data source alias in the tnsnames.ora file present in the same directory as the .exe.
4. data source alias in the tnsnames.ora file present at %TNS_ADMIN% (where %TNS_ADMIN% is an environment variable setting).
5. data source alias in the tnsnames.ora file present at %ORACLE_HOME%\network\admin (where %ORACLE_HOME% is an environment variable setting).

I believe the reason your sample works with Oracle.DataAccess but not with Oracle.ManagedDataAccess is that Windows registry based configuration is not supported for the latter (see documentation) - the ODP.NET installation sets an ORACLE_HOME registry key (HLKM\SOFTWARE\Oracle\Key_NAME\ORACLE_HOME) which is recognized only by the unmanaged part.