There's no best software. Some are useless just in your cases due missing functionality, some others need too much work in your cases and some of them cost a fortune. The rest can be used, but if one of them is already fully familiar to you and is available, it can be a good selection just for you.
Book mockups from scratch are discussed earlier. Here are two cases:
Creating 3D book cover mockups from scratch
Imitate a photo realistic rendering of a book cover
I have given answers to them. I suggested Illustrator. It makes pretty fine stand alone images, but it becomes very laborous, if many different projections or item combinations are needed. Also changing images on surfaces is a challenge.
Complex geometric forms and multi-item layouts need some 3D application. Illustrator's 3D is limited to extruded and revolved shapes and you can have only one 3D object under editing at a time. You cannot for ex. search a good watching projection for a bunch of items by rotating and zooming them at the same time.
In one of the linked cases a book publisher wanted a scriptable system to create semiautomatically new content to their book catalog in their server. Illustrator was out of question in the beginning, no matter how nice was the manually crafted book image.
Affinity Designer can be useful if you have blank 2D images of 3D objects and place only flat images, texts, the background and some shadows.
Illustrator do more. As said, it can extrude and revolve simple 3D items and place images onto surfaces, even onto simple enough curved surfaces. Affinity Designer has no 3D. Image distortions are also limited. For warping an image to fit on curved surface or creating a non-parallel perspective one must jump to Affinity Photo.
Photoshop has one big advantage. There's no problem to change surface images - flat or warped, either go - without tinkering again through the placement details. Only be sure, that pixel dimensions and colors fit. That's possible by using adjustment layers, layer styles and smart objects & -filters. Warping can be smart filter. Photoshop's 3D cannot properly create new models, it only shows existing ones. Thus books and other geometry items are needed as 2D images.
You wondered "How Cover Vault does it?" Their page says, that their mockups are based on Photoshop's Smart layers. In every mockup the geometry is fixed, there are high resolution photos or to 2D rendered 3D models of blank books. Cover vault gives as freebies some mockups. The actual content of that site is a link to commercial mockup collection.
To get the wanted 3D items when needed you should have also a 3D program. One which makes fine photorealistic renderings with placed texture images is a big relief (if you master it). The minimum is the ability to create the needed geometries and export it as vector or high resolution 2D images for coloring and inserting surface images and environment in a 2D graphics program.
An example of shapes which are difficult in Illustrator, but quite easy in 3D software
These aren't straight extrusions nor revolved forms, making these is possible only by drawing them as 2D images in Illustrator. In addition one cannot rotate easily a scene which has more than one 3D object. They must be handled separately in Illustrator. In a 3D program a scene can have as many objects as one wants.
Here's a case which shows placing an image onto a curved surface just as an example of covering a non-rectangular and non-planar form
how do I warp a rectangle to part of umbrella/sphere?
I do not recommend anything without knowing the exact need.