Sure, she's a healer, but there are other healers already in game that seemingly serve a better purposes than her
Not for healing, no. She's the best healer by a massive margin. No other healer can come even close to her numbers.
What happened to her ability to restore an ally's mana? Why was that removed?
To make way for other interesting mechanics. They've said they might do the mechanic again on another champion, but Soraka's new design worked better without it. Besides, she's strong enough that she would need a major nerf if they added the mana restore to her Q, and I can't think of anything I'd be happy nerfing to add the mana restore in. It just doesn't fit her kit anymore.
Also, what about her self- heals?
That was removed so her healing could be strong. When she could heal herself, she just became a battery in lane. If you couldn't completely break her in an all-in, then she quickly restored herself and her ally back to full strength. That potential meant that her actual healing numbers had to be severely limited. She was so safe as a healer that she wasn't allowed to actually be a very good healer. By adding in the health costs and forcing her to put herself at short range to heal over time and to also hit her Q to sustain herself, they added risk to her kit, and that risk means that they could increase her potential W healing by more than 800% (and that's without the higher AP ratio). This was what let her become actually good at healing. If she got easy self-healing back, she'd get her healing numbers gutted again, and I don't want that. I like being a strong healer.