The arrogant men play the numbers game. Most women make the mistake that the man actually likes them but he is just playing the numbers to get his leg over and, if rejected, will just move on to the next one without missing a beat.
On dating sites it is not unusual for such men to contact a 100 or more women in a single hour within a few miles radius - a sizeable percentage of them will end up being bedded by the men as the numbers game is a winner. You won't be asked out by a shy, sensitive guy on a dating site. It ain't going to happen.
The shy, sensitive, caring men fear rejection - rejection is painful and after two or three rejections a lot of men simply give up altogether - and rarely will ask women out, even those they are besotted with. If a woman likes shy and sensitive men then, bluntly, she is going to have to make all the moves.
Women used to know this stuff and, for generations, would manipulate such men so that situations arose where they could get the man, and the marriage, that they want. (Haven't you seen all those 1950's Rom-Coms with Doris Day, Marilyn Monroe and Co?). Modern women have been brought up with a huge sense of entitlement that they just have to breathe and everything they want in life will turn up on their doorstep - it does not work like that. It has never worked like that.
The good news is that, generally, shy and sensitive men are not wanted by women despite what may be posted here - most women want arrogance and dominance. (Don't believe me, just spend a few hours reading female dating profiles.). That means that you have a wonderful selection of lovely men to choose from if - IF - only you can work out that it is you who is going to have to do something about it.
Best of luck. There are simply loads of wonderful men out there who will make fantastic friends, lovers and partners.