There are no requirements to ask about the Rival's motivation. To Flirt, you need to have danced at the Dhaal party and chosen 'Rescue' at the Farrow Facility.
Uncle Max has told his children to claim Victor's probes, or else he'll disinherit them. The non-Rival twin decided to get a 'real' job instead of trying to screw you over.
The Rival seems to have a large, but not unlimited, budget.
1. They have a ship that's larger than yours and a pilot (may or may not also be their current Mercenary).
2. They don't mod themselves extensively, but the Lynx ears and tail seem to be a custom job.
3. They try to buy the Probe from Shekka, but do seem to have a price limit.
4. Johanna claims she's the last Mercenary the Rival can hire because of bad reviews as an employer. Probably doesn't just hire more than one at a time because the Rival is both extremely arrogant and a jackass. And possibly because it's harder to travel with a large party than a small one.
And maybe security concerns: "One has my personal, encrypted key. With that, you can open my deposit box in the bank I own on this station and take control of my company."
Is there anything preventing a hired mercenary turning on the Rival (or you) and taking the key?
As for how they keep getting ahead of you:
Mhen'ga: They trick you by leaving earlier than they implied they would, because they have a pilot and you don't. They also sabotaged your ship, so you have to deal with FIRST before making it to Mhen'ga. No matter what, they get the coordinates before you do and fly off. You have to walk back through the Jungle.
Tarkus: You have to deal with the Stellar Tether, which turns out to be completely unrelated. Also "It shouldn’t be hard for a woman of my resources to keep track of a gnat like you, after all."
I expect this line explains a lot.
Myrellion: The SteeleTech Probe is rather close to a MaxCorp probe. If the planet gets nuked because of your actions during Kara's quest, Dane sends you the next set of coordinates.
Zheng Shi: You fall asleep scanning the system. The 'password' is an attitude check. The Rival may have gotten the answer from Feruze or came by it naturally. Or shot their way in, like you can. I wouldn't be surprised if the Rival is the reason the elevator is broken.
Dhaal: Somehow managed to finagle their way into the party in the serving staff while you were getting an official invitation. Hired a spy (who you get information from later).
That's assuming you beeline toward each Probe and don't stop to do any sidequests along the way.
Canonically, most of the time spent travelling between planets is waiting in line for the Gates. I expect the Rival does a lot of bribing to get to skip ahead.
Having a pilot possibly means the Rival can be more productive than you are while travelling.
Paige reduces travel times by 25%. It's entirely possible the Rival has someone similar on staff. And while you spend a week recruiting Paige, the Rival can spend that week doing research for the next Probe.