If you think it can use air-refueling than you maybe wrong because in air combat a good pilot will shoot down AWACS and Tankers, so there is no need to fight against F-22, which indeed it has lower RCS.

Think about missiles ranges and the maximum ranges are only theoretical. If a missile has 150km range than it has a change to hit an aircraft from maybe 50km. USAF doesn't have long-range missiles like K-172 or R-37 which in tests hit the target at 300km range. K-172 has a maximum theoretical range 1000Km but it is recomanded to use it up to 400Km and against large planes like tankers and awacs.

The range you're quoting there is actually the ferry range. That is the distance an F-22 can fly between friendly airfields. That means no weaponry + external tanks and it is a one way trip.

The combat radius is less then half that and half again typically for US aircraft if you remove the external tanks.

The F-22 has an ingeniously designed unaerodynamic shape which kinda hits the air like a hammer so it has overpowered engines to hit that air hard enough to stay in the air. The problem with that is the fuel consumption. The F-15 has a far superior range then the F-22 due to a more aerodynamic (yet not stealth) design. It is the true air superiority aircraft because it can stay in the air for some time and the Flanker series were designed to compete with that. The F-22a fits into a nice little niche left open by the last generation, where their radars and BVR and fuel capacity and agility mean little. However in order to utilise this gap you need exceptional mission planning, good intel, radar, tanking etc.

In fact you pretty much need flukish luck. You keep on saying "If you cant see them, you cant beat them." while the majority of air to air battles against equals occur either at visual range or by accidental radar contact. This is because equals deploy across the same geographic scale (in other words thinly). And in this environment the LAST thing a paranoid fighter pilot is going to do is turn his radar on and light himself up to every electronic warfare asset within 400km. So basically your F-22A's are going to rely, at best, on occasional AWACS use or maybe an intermittent 'dummy' radar surrounded by allied planes.

The problem is, all these scenarios will eventually play out, as in most battles of equals these days in the air, with pilots turning tail and running for home the moment BVR missiles get fired. Its just not practical or sensible to stick around in that kind of airspace.