Yes! And if I remember, those races actually lasted 24 minutes, right? I hadn't played a game that did that before. I loved the fact that there was an actual endurance/focus element to that race.
Need for Speed: Hot Pursuit or Burnout: Paradise might be the closest to what you're looking for. They're both open-world games, but I don't think they really have that open-world filler that you see a lot of. They both got remastered releases in the last few years.
GRID: I absolutely loved the original Grid (I think it was called Racedriver: Grid in Europe) when it came out.
Project CARS 2 and Assetto Corsa Competizione: A while ago I tried using a PS5 controller on PC and using the gyroscope to steer left and right by tilting the controller. It works well enough when you get used to it. It gives you more granular control than an analog stick. You really can't tilt an analog stick 15 degrees consistently, but you can tilt the controller like that consistently. I'm not saying its as good as a racing wheel, but if you don't have one, it'll at least let you play games that might otherwise need a wheel. I played a decent amount of Project CARS 2 and Assetto Corsa Competizione that way.
Sonic & All-Stars Racing Transformed is a fun kart racing game. If you don't have a Switch and you want something like Mario Kart, you should pick it up. It isn't just a Mario Kart knockoff with Sega characters. Wait no... that's exactly what it is, but it's a good one.
Whoa, I didn't know that. The Dreamcast's 9/9/99 launch date in North America is so memorable, I had no idea that the Playstation came out exactly four years earlier. So there are two birthdays today!
I can't imagine this one getting a PC release, so I'll probably never play it, but I do love seeing a good 3D platformer being released. And a console mascot 3D platformer? Feels like we're in an earlier console generation here.
Right now, a kind of weird one: the Bridget MX, from SGF Devices. It's a 3D printed, all-button controller for fighting games. They don't make that specific model now (it was a very early one), but this is the closest to it: https://sgfdevices.com/products/bridget-pe
At first I thought that not having a joystick would make games kind of boring. Like, too practical, not enjoyable. But no, it's actually fun. Kind of like tapping out notes on a piano. It uses low-profile mechanical keyboard switches, and I have some stiffer, clickier switches on the way right now.
It's meant for fighting games (Street Fighter, etc.), but I've used it for some 2D platformers and it worked great for those, too.
A non-3D-printed, less cheapo one would probably be even more fun to use, but I think I'll stick with this one for now.
There are really no low points in the entire soundtrack. Although, I haven't actually played all of the DLCs, so I might have not even heard all of them.
In case anyone doesn't know, this game is basically Devil May Cry but in the Metal Gear Solid universe.
I'm telling you that because I didn't know that when I bought it. I had played MGS 1 and 2 before buying this game, and it took a couple of hours for me to realize, I don't think I'll be sneaking up on anybody in this one. It's a good Devil May Cry game, though!
This might not be exactly what you're looking for, but Neon White is one of my favorite games of the last few years, and it's on the Switch. I played on PC, but I haven't seen any complaints about the Switch version.
I don't really know if I'd call if a first person shooter. It's more like a first person platformer and you have to shoot some targets before completing the level. Levels are very, very short, and you'll replay them many times to shave a fraction of a second off of your time.
A few super cheap, not-new games I've played that aren't on the main promo banner:
Aragami - $3 - Pretty challenging stealth game. The sequel is also on sale, but I hear it's more action-oriented, less stealth-oriented than the original.
X3: Reunion - $2 - Sid Meier's Mount & Blade, but in spaaaaaaace. (I've only played the sequel, X4. But this one looks kind of similar, and is two dollars.)
Tom Clancy's Rainbow Six - $2.50 - Tactical FPS. Been a while since I've played it. Was definitely cool when it came out.
And if you do run into any problems, the Steam overlay should solve any compatibility problems, if you're OK with launching your games through Steam. I end up doing that with all of my games (including the ones I didn't buy on Steam) since I think its controller mapping UI is the best.