You can absolutely use a diode instead of a tube (i.e. gas laser) and mirrors - or any of the other many ways of producing a laser (doped fibre, solid state laser, semiconductor lasers, etc). The key things are: 1. being able to focus the laser on to your material and 2. your material being opaque at the wavelength of the laser you’re using. Generally for visible-light lasers (most diode lasers), most material that isn’t transparent or reflective to the eye will work. In terms of focussing the beam, most diode lasers also come with a lens that is appropriate to focus the beam over a fairly useful depth of field. So, yeah all that really matters is the power you get out of the laser - which will determine how fast / well you can engrave or cut various materials.
About that… there’s a lot of specification exaggeration going on with laser diodes. Often a ‘10W’ Chinese laser might actually output <2W in usage. Fortunately, equipment exists to measure the actual light output power of laser diodes. I’ve been scouring the internet for about half an hour to see if I could find a list of measured laser diode powers - I’ve defintiely seen one before, maybe on a laser pointer forum. But I can’t find where the hell it is. Anyway, more reputable suppliers will give a more accurate account of the actual power their diodes produce. Expect to pay around £100 from a cheap Chinese supplier for a diode that can actually produce 4-5W - which is a good amount for a decent, if fairly slow, laser engraver like you’re describing. Cutting through darker shades of acrylic up to a couple of mm thick will be possible too. For the same power from a reputable supplier who has actually measured the power, it might be £250+. Unfortunately, I have no idea what power the particular laser you’ve linked might actually produce.