Breastfeeding vs Pumping

This is a constant debate I have had in my mind pre-baby Gemma, and was something I had to choose between with Elle when she was a baby. I keep reassuring myself that both provide the baby with the best food possible for their growing bodies and minds, breastmilk. However, for some reason I always feel guilty if I am unable to directly breastfeed and instead choose to exclusively pump/bottle feed.
With Elle I tried breastfeeding at the hospital but never got her to really latch so I came home and started to pump. Part of me hoped this would be different.
Pumping was fine but it is so inconvenient and time consuming. Because so many of my friends had babies the same time I had Elle this was very noticible. We would go to the mall or zoo and if their baby got hungry they would just "throw the baby on the boob" and keep going with life. I however had to have bottles packed in the cooler, find a place to warm them up and even find places (usually bathrooms) to pump myself so I didn't start to leak. I always found myself seperating from everyone. People would come visit and same situation, I always would have to leave the room to go sit somewhere alone and pump. It drains on you, especially when you are still trying to be a mom and get that all figured out.
At the hospital this time, I again tried to breast feed. Gemma actually latched and it wasn't perfect but it was working out. I was shocked. Well as the hours passed, she kept trying to breastfeed but it started to get harder and harder. She was very aggressive and would latch, let go and then latch again...problem being it would hurt me so bad. I tried toughing it out but eventually the pain was too painful and I couldn't do it. She actually was so aggressive that at one point she actually got a hold of my boob, not my nipple, and pulled a chunk of skin off.
So once again I came to this debate, what do I do? Well the pain wasn't something I could really tough through so we came to pumping again. I pumped in the hospital and it wasn't ideal, still hurt a bit because at this point my nipples were raw and bleeding a bit. But the lactation consultant reassured me this will get better once they heal. I didn't get much colostrom at first, which was frustrating but I have high hopes I will have a lot of milk, like I did with Elle, eventually.
Well we are about 2 days into this pumping and my milk is starting to come in and my supply is definitely starting to grow. It is a big reassurance to see the milk in the bottles. It motivates me to keep going and to see that I am doing something that makes a difference. I am still annoyed by having to stop what I am doing and pump ever 2-3 hours but deep down I know what I am doing is right for Gemma. She still is getting breastmilk, but she is just getting it in a bottle. A positive of that is Chad and Ellery can help feed her this way and bond with her more. I know pumping is the best choice for our situation, but it still hurts a bit when people see me feeding her a bottle and say "oh I thought you were breastfeeding..." Then I have to have that awkward conversation that I am breastfeeding just through pumping. It's awkward because I feel like I have to justify what I am doing to them. Hopefully, I can not let those conversions get to me and I can just keep pumping strong.