I personally find characters who are hot messes in various ways to be much more interesting and compelling than people who are highly functional across the board. Personal dysfunction can become yet another obstacle to overcome. I like stories where not all the obstacles to success are external, in other words.
And who in the heck doesn't have (or hasn't had) issues at some point or another in their life? It can be darned comforting to see people with serious problems overcoming adversity and to see evidence that we ourselves are not alone.
There are some people who don't like them, but from what I've seen, even in genre fantasy (which has been accused of having more than its fair share of overpowered, wish fulfillment characters), flawed characters, and characters with lots of emotional baggage, abound.
I think something that can make a character lose sympathy, however, is if they are not functioning as well as they reasonably can be at a given time and place. If a character makes poor choices because he or she lacks information, or has problem that logically leads them to screw up, then it usually works fine. But if they come off as too stupid to live, or are making mistakes that seem out of character, or not learning at all from their mistakes (unless that's the flaw you're illustrating), or if they are overcome with paralysis for too long, then some readers will likely disengage from them.
Another thing that can be frustrating is if the character is static. If he or she doesn't overcome at least some of their problems, or find a better way of coping with them, by story's end, it can frustrate some readers. Though I can think of at least one fantasy series where the very flawed characters don't really grow or change at all (they seem like they are for a bit, but then they don't), and yet it worked somehow.
The main thing is to make your characters the people they need to be for the story you are trying to tell. Not everyone will like your approach, but if your write well and from the heart, some likely will.
Some books (just off the top of my head) that have done well with very damaged characters:
Trainspotting
The Hotel New Hampshire (and most anything else by John Irving)
The Accidental Tourist by Anne Tyler
The Prince of Tides by Pat Conroy
Almost anything by Joe Abercrombie
A Song of Ice and Fire series by GRRM
The Color Purple (and its sequels) by Alice Walker
Many of CJ Cherryh's SF books
The Farseer and Fool books by Robin Hobb (plus others by this author too)