First of all I really don't like when people call cats "dog-like", it's a pet peeve of mine and a big rant I can go into but I'll just do a mini version here.
I.e a cat that will play lots with you, is happy to see you and is there at the door when you come home and needs lots of interaction isn't dog like. It's a social cat. Cats are cats, not dogs, and if you want a dog you should probably get a dog but if you want a cat all cats can be social fun and active, especially if you raise them from kittenhood.
Cats love conditionally, so that you get back what you put in. Because they can be independent and are very adaptable, cats that don't get much interaction when young can become the stereotypical "aloof" cat, just because not getting interaction and being left alone is what they've always experienced so they adapt to that. Changing a cat when it becomes older is possible but can take a lot of time and sometimes won't work at all.
Pouring love and play and attention on a kitten or a young cat pretty much always works
No breed of cat is inherintly superiour to others. Cats are all individuals and can be very different. I own a Bengal cat which is originally a hybrid breed like the Savannahs.
Both Bengals and Savannahs are very active and intelligent cats. This sounds like ideal to a lot of people but they still don't quite know what they're getting themselves into. These cats need a lot of time and attention and quality play time. Not just when they're new and you're still hyped up about their addition to your household but every single day for many many years (if you're lucky). You being tired, your living situation changing won't change the needs of the cat.
I live alone, work fulltime and have a single bengal cat (where I rent wont let me have more than one cat). Not ideal but I get by and keep Nikita happy by playing intensely with her for over 2 hours every single day. This has cut into my social life but I don't regret that at all because I love her and getting to share my life with an amazing creature more than makes up for it.
Anyway if I were you I would first of all really think about exactly why you want a pet and what level of commitment you're prepared to give, then find a local shelter and talk to the people there (unless you're set on wanting a certain type of breed like for example the Savannah, in which case you should learn as much as you can about the breed, where it came from and history to get a feel for what to expect). Anyway the people at the shelter should be able to advice you and they usually know the cats/kittens they have and can help get a personality match.
And to answer your original question, purebred cats are no "better" than random bred cats. What you can get from purebreds is statistically higher odds at ending up with a certain personality type. They can all make absolutely fantastic pets, but like I said earlier you tend to get back what you put in when it comes to cats. A cat won't automatically become a fantastic pet, they need attention and socialisation and play to bloom.
Also if you do want a purebred cat, go visit a breeder and see the cats and play with them if you can to get a feel for what they're like. Your idea of what an "active" cat is might be quite different from people used to the breed or who have otherwise very active cats. That happens all the time with me and Nikita. My friends usually underestimate her energy level when I tell them about her and then think she's completely insane (but in a good way if they're cat people) when they actually meet her.