Let's start the long list:
The paddock:
Reworking career mode
Career mode is just plain race after race after race. To make it unique you need to introduce off-track things, such as interviews, teammate battles, etc. After race cutscenes need to vary, rather than being 10 sec sadface for P4-22 and 10 second happiness for P1-3.
Multiplayer career
Not just a co-op career. Let more than 2 people fight for the entire season!
Editors
It doesn't take a lot to give us helmet/car/even database editor, if you release a game so close to the season end. Let people make some work for you!
Unique game modes
F1 is more than just drivers. Introduce more game modes, like the Steward mode, where YOU are making the decisions about penalties and SC instead of messy AI. Of course it can be exploited, but you can add ranks for stewards. If they drop beyond certain point, they can't be stewards for some time.
Spectator Mode from the very beginning will give the power to Online Leagues. Commentators will be free to choose who to follow or just have random/not so random footage to work with. Sticking to 1 car is not the best idea for them.
Team Manager/Engineer mode gives tycoon people something to play with. Finances and other management things will look cool and Being the race engineer and communicating with drivers while having the telemetry will be interesting as well.
And imagine the combination of it all in online! The organised league can have A LOT of different tasks for different people! 22 pilots, 11 Engineers, 1-2 stewards, 2 commentators...
Theatre Mode
As much as I hate CoD, their Theatre mode is so awesome! Wathcing race with different stats and multiple cameras would be a fantastic thing for commentators/spectators/viewers. Save the replay and send it to others to watch ingame! Edit the raw footage and save it to your hard drive! Add the commentary on the fly to include to the recording!
The gameplay:
Practice
Re-introduce FP1-2, if players need it, use the toggle. Mid-season testing with objectives give career mode even more stuff to play with. Give players the chance to drive without competitive racing for once, it can be interesting too!
AI
New AI. For sure. Return lap blocking in quali, cowards in races, no-collision races... This makes the racing boring - AI being either stupid or predictable. You can always defend by late-braking, AI will always stay behind, being afraid to risk and attack. Pit-stop will take ages, if a few cars will enter pit-lane within a few seconds from each other. They don't even spin out! Race engineer should not tell you "There are a lot of acrs on track today", instead he should tell you who is on different tyres and other important stuff. He should give you mid-race addvice, cause he is the one who sees the whole picture!
Penalty system
Enough said. It is horrible. Even improve it A LOT, or let players handle the issue, as suggested in "Unique Game Modes"
Damage
First of all, how cheap do you need to be to avoid animating the broken car removal. At least use the pit crew quali animation, when they push the car away. Also, let they debris stay for some time, make drivers work their way around, giving them a chance to have a puncture. Send the SC out and when no one is around, remove the debris.
Add technical failures, which will force the car to retire if damaged (drivetrain, engine, etc.), introduce random failures (not just KERS/DRS failures) based on teams IRL performance. IF players do not want to randomly retire, they can turn mechanical failures off in the damage menu, just add the slider.
More Than Just A Race
F1 is not just a race from green to checkered. Small stuff like a warm laps are important as well. Introduce manual warm-up laps, so drivers will be responsible for tyre performance during the start.
Add jumpstarts to make players actually react to the start.
Add manual pit stops to make players work even there to get the car in and out of the pits in the least amount of time possible without getting a penalty.
Add the after race return lap to let the players celebrate their victory! And for those aggressive drivers, make them drive back to pits with that fuel, which is left on-board. Fuel management to the max!
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I have so more minor wishes, but here ends my list. Hopefully, one day I will see a perfect F1 game.