I've heard the "you're to independent" one a lot over the years - sometimes as a break up excuse, sometimes from friends (male and female) as the explanation for my singleness!
I think there are some men who like to be in the limelight all the time, and feel threatened if their partner in any way or at any time outshines or challenges them. These men really want a housekeeper/prostitute, and run a mile if you step outside the scripted role. For these, I don't think there's any point in trying to change yourself - it's a one way road to hell. You're no longer true to yourself, and the more independent you were to start with, the harder and more frustrating it will be.
On the other hand, the longer you've been alone, the more independent you get by force of necessity. Of course you can unblock the drains, change a plug, use a drill and fix the fuses - if you didn't these things, they wouldn't get done. Most of us can't afford to pay for "a little man" to come and hang a picture for us - and the more of this stuff we do, the better we get at it. This is traditional "man territory" - and it can be diplomatic to feign ineptitude with these things with some men to give them a chance to play "cave man"!
Personally, I'm totally useless at this - I'm so used to doing stuff myself, it doesn't even occur to me to ask a man to do it - though if he's around I might (nicely) ask him to hand me the screwdrivers while I'm up the ladder. Then I get thoroughly ticked off when he runs off with a pretty little thing dressed all in lace with a bow on her head. Hmmm. Story of my life. Not good. I understand the theory - just can't get it into practice!