Starting a sentence with 'Well, ...'

Well is a Discourse Marker. What it means, in context, is dependent on how it's pronounced, especially its length and tone. There are dozens of possibilities, some neutral, some positive, some negative, some ironic, et complex cetera. Anyone can construct a comedy routine out of different pronunciations of Well, ... construed as dialog.

In writing, since it refers to sound and not words, it means whatever the writer intended it to sound like. Consult your mind's ear.

See also Uh.., Oh.., Hmm.., etc.

Well¹, I believe there is a whole range of possibilities, and none predominates; witness to this point of view, the OALD definition for this exclamation provides nine meanings; none of them is specifically this particular meaning of negative opinion in which smugness can be a nuance, and I believe it could be added, as well as another one.

¹ _{See "5" below}

(OALD)
1 used to express surprise, anger or relief
♦ Well, well—I would never have guessed it! ▲ Well, really! What a thing to say! ▲ Well, thank goodness that's over!

2 used to show that you accept that something cannot be changed
♦ Well, it can't be helped. ▲ ‘We lost.’ ‘Oh, well. Better luck next time.’

3 ​ used to agree to something, rather unwillingly
♦ Well, I suppose I could fit you in at 3.45. Oh, very well, then, if you insist.

​4 used when continuing a conversation after a break
♦ Well, as I was saying…

5 ​ used to say that something is uncertain
♦ ‘Do you want to come?’ ‘Well, I'm not sure.’

​6 used to show that you are waiting for somebody to say something
♦ Well? Are you going to tell us or not?
​
7 used to mark the end of a conversation
♦ Well, I'd better be going now.

8 ​used when you are stopping to consider your next words
♦ I think it happened, well, towards the end of last summer.
​
9 used when you want to correct or change something that you have just said
♦ There were thousands of people there—well, hundreds, anyway.

((user LPH)
10 used to make reprobative remarks (with a possible nuance of a holier than thou attitude)
♦ Well, well! Look who's here asking for help when they've been telling you forever haughtily they had nothing to learn from you!
♦ Well, when you are not happy with your work you don't make that the reason of your asking for leniency.

11 used to remind someone that they overlooked something
♦ Well? I just told you that a while ago…)

The possibility considered (smugness, holier than thou attitude) is not to be excluded, but in at least a significant part of the cases that can be collected, one must expect that "well" is used as a term expressing partial agreement or incertitude (3, 5), as a means of introducing something of interest (4), as the introduction of words meant to remind someone of something (11), and also, if not as often, to render any of the other attitudes that this word connotes.