I was wondering if there are any huge cacti like these:
(I believe they're called Saguaro or something, but I guess similar cacti are fine too) that one can see (in the winter) somewhere along the following route?
San Diego - Palm Springs - Phoenix - Sedona - Tusayan - Page - Panguitch - Springdale - Las Vegas - Badwater - Los Angeles
I'm aware about Saguaro National Park, but it's too far from Phoenix.
Saguaro cactus are native specifically to the Sonoran desert. This photo outlines where the Sonoran desert is located.
According to this TripAdvisor thread, you will not see them 20 minutes north of Phoenix. They seem to stop growing above 3,000 feet above sea level. You should be able to see them south of Sedona, along I-17 interstate highway.
You can find Saguaro in South Mountain Park on the south side of Phoenix. Here's a screenshot from Google Street View showing a couple:
Your segment from Palm Springs to Phoenix will probably be along I-10, which passes through Blythe at the Colorado River between Yuma and Lake Havasu City. There will be scattered saguaro all along that stretch of road and more densely populated patches just off the highway, especially to the south-east of Blythe in southern Arizona. When I lived in Phoenix 50 years ago, the town (now "city") was lousy with saguaro, cholla, barrel cactus, and prickly pear characteristic of the Sonoran Desert. The stretch of I-17 just north of Tucson is always a great place to see loads of them, but that is not part of your stated route. I also suspect the Colorado River valley between Blythe & Lake Havasu City will not be part of your itinerary; if it were, you'd see all you'd want and more...
Leaving the Phoenix area to the east along Highway 60, there are many many Saguaro between Apache Junction and Superior.
Leaving the Phoenix area to the south along the I-10, once you leave the developed area (not too far south of Chandler), you again can see many of them, and can continue to see many of them all the way until you hit the border of New Mexico.
Leaving the Phoenix area to the west along the I-10 will have some visible near the interstate, though not in the profusion to the east or south of Phoenix.
Leaving the Phoenix area to the north along the I-17 will briefly have some visible, but the elevation rapidly climbs several thousand feet out of the valley, and you won't see many after that.
Your chosen route seems to be too far north to enounter Saguaro.
There is also Ethel M Botanical Cactus Garden 10 miles away from downtown Las Vegas. According to the pictures from Google Maps, there are some big cacti there (I don't know whether they are Saguaro or not).
Those cacti are not quite "in the wild", but the question as I stated it does not specify whether I'm looking specifically for cacti in the wild. (In fact I am, but nevertheless this answer may be useful to someone.)
The drive on I-17 between Sedona and Phoenix has wonderful views of saguaros from a car, but not places to hike among them. You should definitely stop at the Desert Botanical Garden in Phoenix; they have great specimens and information. The South Mountain Park and Preserve in SW Phoenix has some great hikes with plenty of saguaros.
You can't go to Phoenix without seeing saguaros!
This isn't answering exactly what you asked, but if you are travelling through Palm Springs, and interested in the unique flora of the American southwest, you should make a side trip to Joshua Tree National Park, home to the a desert plant just as iconic and even more restricted in range than the saguaro.
