Your manager is short sighted, the best path goes through both 'options'.
If you are going to be a 'real' tester, then you will need to master 'manual' testing first. This is the best way to test 'new' things, and pretty much the only way to get tests which then can be automated.
'Automated' testing is best utilized for 'regression', that is, making sure that no unintentional changes were introduced along with the 'new' or 'changed' function(s). It is an additional test skill, to translate an existing (manual) test case to whatever automation is available and appropriate. Thus, to do it 'right', you would need to be a competent manual tester to develop or at least completely understand the manual tests to be automated, as well skilled in the automation techniques.
It is claimed that there are those who 'cannot' do manual testing; they just convert existing test cases to automation. In my opinion, these are not 'testers'; they are at best 'translators'...