Here is what will happen with visible pickups :
Let's suppose there are currently 3 pickups on the portion of the track :
2) normal missile
3) normal missile
The first car arrives ; as he's first, he gets to choose between turbo or normal missile. The choice is obvious here : turbo.
Then the second and third car arrive. They only have normal missile left.
Result : the first one is even more advantaged than before ; and anytime there will be a turbo on the track, the first one has higher probability to take it than the other racers, resulting in an unfair advantage for the first one.
Actually, such system is implemented in Quantum Rush. When I was in the closed beta, I could see that the first one on the first laps was very often the first one in the end of the race (assuming he was good enough), simply because he had more boost bonuses than the others (also, there was no catch up mechanism).
That's why I disagree with the visible pickups
Now about the semi randomized stuff ; I believe what you describe is something like "Pick a random pickup from a pool of pickups that you haven't picked yet". Once the pool is empty, you reinitialize it. So assuming there are only 3 pickups in the game called A, B, C, you could have this kind of order :
A, C, B (reinitialization of the pool), B, A ,C (reinit of the pool), ...
(actually you didn't state about the reinitialization ; but after all pick ups have been picked up, you have to do something, or else your system will be effective only for the 3-4 first laps ; that's why I assumed that you were just reinitializing, which is the most simple way to handle it)
The big problem of this approach is this : Let's assume there are n different pickups in the game, and let's assume you have taken a particular pickup (let's say an armor), but a couple of seconds after you need it again.
Let's suppose we are in the worst case possible, which is that the armor was the first of the pool of pick-ups.
Then you have to pick (n-1) pickups before the pool is being reinitialized. And then, in average, you will have to take a little less than n/2 pickups (didn't make the computation but it should be lower than n/2 for sure) to pick the armor.
In the end, you would have to take a little bit less than 3n/2 - 1 pickups before you can have the armor again. Now let's suppose it's Rollcage 1 with roughly 10 pickups. That would mean you would have to wait in average 14 picks before you take again the armor. If you want the min-max, it's between 10 and 20.
I can't think of a more frustrating system to be honest