An economy system or unlock progression could be a good way to motivate this.
Here, you'd have a wide variety of parts - weapon & defensive tools with differing style & effectiveness, or vehicles with more or different arrangements of weapon & defensive system space, or additional characteristics like cooling rate perks. You'd gate some of these parts so that the player can't use them all in their very first race.
The gate could be a cost you need to pay or point threshold you need to hit to unlock a part type for use, or even a modest installation fee you pay every time you want to equip a part.
Killing all of the other players gets you a win and lets you advance to the next race, but is not the most effective way to earn the points or currency used to get new gear.
Instead, every time the player completes a lap (or even better, at every checkpoint) they earn a batch of this currency.
The value can increase from lap to lap (so I want to get to the late laps for maximum earning), or even award a bonus for the number of competitors still in the race (fictionally: your audience likes watching a more dramatic, contested race) - this encourages the player to play like a race and get ahead early, rather than thin the competition first before picking up an easy few laps with fewer opponents to fight.
Winning the race by crossing the finish line can grant the biggest pot of currency, making it a clear carrot to draw players forward.
So players who want to race with better and better equipment (or at least, more customized to their style) want to get far in the race to build up their points/currency for the next race.
Of course, at some point you'll run out of new & interesting gear to offer, so you can keep the motivation going with things like...
"Prestige" levels the player can advertise on their name/profile for others to see ("I've earned enough points to unlock everything three times over!"
Leaderboards for the top-scoring/earning players today / this week / this month
Vanity customizations purchased / unlocked with the same points or currency (colour changing, decals)