Mask

What makes a bike difficult are all of the enclosed background areas, like between the frame and spokes. In older versions of Photoshop, like the Creative Suite versions you said you had used, this was an almost completely manual process. Today, this is an area that’s evolving rapidly. Remove Background is a relatively new feature and the early versions were disappointing, missing a lot of areas. In recent months, Adobe has upgraded the AI model so that it’s a lot more accurate and requiring much less manual cleanup, so you might try that. 

Below is a video showing the improvement I’m talking about, and it even uses two different bikes as examples. When he switches Select Subject to the Cloud (Detailed Results) option, the selection of the bike is dramatically improved and much more detailed. If you don’t get similar results, you might have to do it in the Photoshop public beta like he is showing.

There are a bazillion tutorials online about masking.

https://helpx.adobe.com/photoshop/using/masking-layers.html

https://www.adobe.com/products/photoshop/remove-background.html

Instead of remove background, make your selection then use Layer > Layer Mask > Reveal Selection. An editable mask will be added, which you can fine tune if required. You can also put a new layer below your masked layer if you do not want the image to be transparent.
Dave