Fuck no. Hormones (and most other injections) has way more impact / risk / effect than a tiny cut.
I cut myself several times a month, badly a few times year. (too lazy to wear gloves when doing yard work, accidents etc). I know what a cut is / does. No idea what putting crap in my body will do.
You cut yourself badly a few times a year? To each his own, but that sounds fucking crazy to me. I manage to live my life both without artificial hormones and without non-emergency invasive surgery and I haven't had as much a a paper cut in as long as I can remember, probably over 10 years. Also I am not pregnant or had any pregnancy scares.
I should have defined "badly" as many people probably think badly == stitches. badly to means means enough to leave scar, needs cleaning & covering (bandaide).
As far as invasive surgeries (and birth control) goes, it's really not that invasive if it can be performed in 15 minutes without general anesthetic by a regular doctor. Even IUDs require more time than that.
Would you still be okay with the hormones if they caused you to gain 15 lbs of fat, lose your libido and render your SO unattractive to you? It's not as simple as swallowing a pill every morning and getting on with your life -- hormonal treatments have very real and sometimes significant side effects.
What you are describing is a possible effect, but there is no way that is the median side effect of standard oral birth control.
This type of claim always comes up whenever there is any story about new birth control. The effects of the hormones cannot possibly be worse than just not having sex since you always have the option to not have sex or always use condoms. I simply cannot believe that it is so bad considering literally every girl I know, even the ones without boyfriends and who aren't having sex or planning to have sex in the immediate future. Everyone I know in person talks just as much about how it clears up their acne as the negatives that are always posted in the comments here.
I was on depo provera for over 3 years. It destroyed my sex drive, nearly caused the end of my relationship, gave me terrible recurring headaches and mood swings. I came off it, and after 18 months my body was still all over the place; I ended up taking different hormone-based contraceptives to try and re-jig my system.
Few months after all that I became pregnant with twins but miscarried at 8 weeks. I know that logically miscarriage is 'normal' and common, but a tiny part of me can't help but think that screwing with my body + hormones was the cause there.