As a matter of fact, no. I have that other programmer that doesn't have DTR brought out to a header but it does have that line brought to a row of holes on the board where a header can be added. I have done that and no, it does not work on my office computer.
Where I am is that my laptop and office computer works fine when programming ESP development boards, i.e. where the FT chip is on board. They also both work when using that programmer that does the USB-to-EN/IO0 decoding.
My laptop works fine programming any board, including those of my design, that have the NPN BJT interface circuitry on board but my office computer does not and it doesn't matter which programmer I am using.