I'm going to set aside my Sirius/Remus ship feelings on this topic. From what I gathered, Remus was so used to being shunned and rejected that he didn't feel he deserved love, and rather than "drag Tonks down with him" he would push her away. Tonks had to consistently prove that she was all in, essentially chasing after Remus. On the surface, it looked like Tonks loved him more since Remus was all but ready to abandon her at the first chance even to help Harry's mission. But I think Remus's insecurity proved that he loved Tonks enough to truly believe she deserved better and wanted the best for her.

It's sad to see how they both ended up, with him running off to fight in the war and with Tonks leaving Teddy to be with Remus. Tonks having already lost her father and how down she was throughout the 6th book over Remus' rejection, I don't think she would've coped well living without him. Perhaps like George Weasley, unable to produce a Patronus for the rest of her life.

As for parental skills? I think this depends on if Remus and Tonks's relationship would've lasted. There's only so much self-sabotage that one can take without the other having long-lasting effects of sticking through it. But I think they could've worked it out with the right circumstances. Since after Harry confronted Remus over his poor decision, Remus was proud to have a son. He could've eventually learned that he is also deserving love and be able to share it. Even if they didn't last, they could've co-parented just fine. But sadly, Remus was always going to end up on the battlefield, he had his friends to avenge and family to protect, and Tonks was ready to go down with her true love.