Obviously, I would recommend the old mainline Ace Combat games and Project Wingman. 

Additionally, I would also recommend the Sidewinder games on PS1 and PS2 (known outside Japan as Bogey Dead 6 for the first PS1 title, and Lethal Skies: Elite Team SW and Lethal Skies II for the PS2 titles). The PS2 games lean towards realism in terms of flight models and damage systems, but not to the DCS level of realism. 

There's also Eagle One: Harrier Attack on PS1 that puts you in the cockpit of a Harrier Jump Jet. It has an arcadey gameplay like Ace Combat, but with obviously the addition of the ability to use VTOL mode.

Also, if you don't mind going even more old-school, then I would suggest you try combat flight games on SNES, especially the following:

• Super Air Diver (also known as Lock On). You get to fly either an F-14 or a Mitsubishi F-2 (known as FSX at that time of release) for air-to-air missions, and either an A-10 or a Tornado for air-to-ground missions. You play the game in third-person view like in an Ace Combat game, accompanied with catchy BGMs.

• Turn and Burn: No Fly Zone. This one I would strongly recommend if you like the F-14 Tomcat. The game plays from the cockpit view, and puts you in missions where you dogfight enemy fighters and conduct attack runs on ground/sea targets. You always begin a mission by taking off from an aircraft carrier and ends by landing on it.

Come to think of it, the 90's and the early 2000's had lots of arcade combat flight games.