I think studio flash (either monolights, or box-and-cable system models) is easier to work with than continuous lights. I linked to that one Adorama Learning Center video just so you could see how three inexpensive lights and softboxes can be used to light up a shooting area in a "broadly-lighted" way that allows a squirrelly subject that doesn't take directions very well, so that there's a lot of posing flexibility. Flash freezes subject motion, and eliminates camera shake, and it gives a LOT of exposure power, with each flash. Continuous lighting the way Ms Lackey does it is these days, with a D800 or D810 and often times, pretty high ISO levels and wider f/stops than one might use for pets. She has Nikon's best cameras, and their best lenses; for a D800 user, ISO 2,500 at f/2.8 is not a worrisome issue; with a couple of 150 Watt-second monolights, you could shoot at f/8 at ISO 250, and get plenty of depth of field.
Pets and people are different things too...with people, we focus on the EYES, very strongly...with pets, they are usually covered in fur, and so if there is not plenty of depth of field, the animal looks, well, simply out of focus in many poses...that looks bad, in my opinion. Studio flash units that have 150 Watt quartz-halogen modeling lights give plenty of light to see by and to focus by. I have lights that use 250 watt quartz, 150 Watt quartz, 100-Watt quartz, and 3 x 25 Watt incandescent modeling lights. Many other brands are going to use moderate-power bulbs, like 100 Watt incandescent modeling lights. Those are mainly to help see, to focus, and to guide you in placing and adjusting the lights. Flash gives a LOT more "power", for a LOT less money. A lot,lot less. With less heat as well.
I have some ideas on how pet photos would most efficiently be shot using two, or three, studio flashes and some simple modifiers. You do not need the highest-grade lights, but you do need a minimum of a few simple things to do decent studio lighting. Not "a lot", but "some" stuff. A lot of videos show continuous lights because so,so many beginners are afraid of flash, and therefore, the retailers SELL loads of continuous lighting kits like those in the video. My impression is that very frequently, later on, when the buyers get more serious, they end up replacing the continuous lights with flash systems.