Shabbat Shalom OP.
The SEGA Genesis holds various console games that have arcade philosophy.
Jewel Master, Midnight Resistance, and Atomic Runner are run and gun games that gets you out of the typical Contra and Metal Slug sphere.
There is Splatterhouse 2, and if you haven't played through Splatterhouse 1 then that is a must play first before the sequel since it's a continuation of the story.
Outrun 2019 and the Road Rash trilogy are great racing games akin to Super Hang-On. Then there's Skitchin' in which you utilize traffic to gain speed and catapult you past your rivals.
For some shoot em ups there is Elemental Master in which you can select a few stages and gain new magical shots. Bio-Hazard Battle where everything is an organism. Then Wings of Wor which you're an angel to strike down evil.
For side scrolling action games I highly recommend the Shinobi trilogy, Revenge, Shadow Dancer, and Return of the Ninja Master. ESWAT: City Under Siege is also great as well, and you eventually obtain an exosuit.
There is a gallery shooter called Dynamite Duke which is fairly solid. There are bosses at the end of the stages which you can end up getting into fist fights over!
Mazel tov OP!

>>11800464
Depends on what you're feeling.
Side scrolling action: Jewel Master
Shoot Em Up: Wings of Wor
Racing: Road Rash
RPG: Shadowrun
JRPG: Sword of Vermillion
Beat em Up: Alien Storm
Platformer: Ristar
Vehicle Combat: Urban Strike
Action Adventure: Beyond Oasis
Run and Gun: Contra: Hard Corps
Mecha: Battletech: Armored Combat
Puzzle: Columns