Eccleston's Doctor seemed to be less obviously likeable than Tennant and Smith's. Not to say UNlikeable, but one that could take things more seriously if the situation demanded to - and in the first half of that series, he was rather hard-edged and darker, but mellowed following the whole Are You My Mommy ordeal. Part of that characterization, of course, is embelished in the fact that he's the first post-Time War Doctor, which makes sense of that darkness in him.
That's almost totally not the read I have of Eccleston's Doctor.
My "elevator pitch" read of Eccleston is "Davison with PTSD." He's a bit lost like Davison's Doctor was, and he's impotent like Davison's Doctor was, but he's hesitant in a way that Davison's Doctor wasn't. When I call Eccleston "impotent," it's for the "Coward, every time" speech, which is so contrary to who the Doctor is that I'm baffled by RTD's misunderstanding of the character. (The Doctor who destroyed Skaro and Gallifrey is anything but a coward.) And I don't find Eccleston particularly unfriendly -- except maybe in regards to Mickey, to whom he's a total asshole, but that's not just Eccleston, that's also Tennant.
In short, then, I don't see Eccelston at all the way you do.