A proper bike pump should be able to do both or will likely just do Presta valves. I use adaptors for emergency situations because I want a pump that works well for my tires and is easy to use. Having to potentially unscrew the cap, then adaptor, loosen the valve on the tube, screw back on the adaptor, pump, unscrew the adaptor again and then tighten the valve on the tube and screw the adaptor on again and probably put a cap on that, is a huge pain. I would rather just loosen the valve, pump tighten valve and done.
I will also note that using an adaptor can unscrew your valve cores so if you are using them regularly you will want to have a valve core tool as well to make sure things are tight. It is a good tool to have anyway but way more needed with regular adaptor use.
I would get a
JoeBlow Sport III and not worry. They are not expensive, very reliable and you can get parts for it if needed be. It has a presta side and a schrader side so you can do all valves (including Dunlop. We have sold them for years now and haven't had any issues and it is a quite easy pump to use, we use it as a shop pump as well. It also comes with a ball and bladder needle so you can pump balls and pool toys and such.