"Open Water" is the full course, unless I'm misunderstanding your post.
Look, there are a lot of crazy people on this website who do a lot of crazy things to dive cheaper and deeper while cutting any possible corners, middle men, and extra steps. They're all telling you to take the darn class!
The class teaches you things you wouldn't pick up otherwise. It is possible to teach yourself to dive, but very hard to teach yourself to dive PROPERLY and SAFELY. There are a lot of things you don't know you need to know. A lot of bad habits you can form without an instructor catching them early. Maybe you could get away without taking a class if you had an experienced mentor willing to help you, but they would probably just tell you to take the class anyway. The class is generally pretty easy and fun. Buying gear, getting on dive boats, getting air fills, getting gear serviced, or doing any further training down the line will be very difficult without the certification card. Nice thing to throw on a resume too.
Regarding gear: you certainly could buy your own gear to use for the class and avoid the rental fees, that's sensible. Counterpoint would be that you will learn a lot about what you actually want/need in gear from your class experience and instructor. Everyone has some equipment they regret buying early on. For example, I wish I bought a backplate+wing instead of an expensive jacket BCD similar to the one you want. If you plan to dive frequently and locally you're right that it's smart to buy a full set of gear sooner rather than spend money on rentals. Just do a lot of research first. Look at what experienced divers in your area are using, get their advice on equipment choices. Preferably people who aren't actively trying to sell you things. Local dive clubs would be the place to start.
Regarding the compressor: shop air compressors run something like 250 psi. Most SCUBA tanks use 2400-3500 psi. 250psi in a scuba tank is basically empty. SCUBA compressors have elaborate filters to dry and purify the air. SCUBA air must be extremely pure to be safe for you and your gear. Any contamination in the tank (like oil from the compressor or water from condensation) can be a huge problem. Compressors can pump Carbon Monoxide into the tank either by sucking in vehicle/furnace exhaust in the atmosphere, or by producing it internally when an overheated piston burns small amounts of oil. Breathing a tiny fraction of vaporized oil or carbon monoxide at sea level is survivable, but when you're breathing it underwater at pressure you're actually breathing A MUCH HIGHER CONCENTRATION of the contamination per breath due to the ambient pressure. Not to mention how it will harm your equipment (eg. oil degrading seals, moisture rusting tanks from the inside). Similarly, a headache/blurred vision or even passing out from contaminated air on land is survivable, underwater it often means SWIFT DEATH.
If you do insist on ignoring all of the people telling you to take the class, at least learn to never hold your breath (especially while ascending) so you don't overexpand your lungs and die. Pulmonary embolism can happen in surprisingly shallow water.
I hope I did an okay job explaining WHY people are telling you "no" so emphatically.