For continuous duty on the compressor end you need a rotary screw rather than a piston compressor; you've probably seen road crews towing them behind the truck. For continuous tool use what you need is a compressor which provides a higher CFM than the tool requires. (And technically an air set to inject oil into the line, because if you're stopping to lubricate it manually the tool isn't in continuous use.)
The first problem you're running into is that portable compressors provide very small amount of air. A typical six gallon pancake provides about 2.5 CFM, which suffices to run staplers and finish guns all day. Anything using more air like a framing gun or your scaler and you'll be waiting on the compressor. They can still be useful with large tools you use intermittently.
The second problem you'll encounter is physics. Enthusiastic horsepower claims aside you can only draw a finite amount of power from a standard outlet. Ingersoll Rand's Garage Mate is a popular portable compressor retailing around $600, which gets you a whopping 5.5 CFM. (Portable is a bit of a stretch too, since it has a twenty gallon tank.) That's about all you're getting from a regular outlet.
Unless you're running a large assortment of tools it's probably more economical to stick with corded electric ones if you need to use them continuously.