Having a soul doesn't mean you're good, but not having a soul means you cannot be good (though you may act good sometimes). Actually, it's not a matter of good or evil per se, but rather of conscience - an inner sense of right and wrong.
Some people are sociopaths, so having a soul doesn't necessarily guarantee a conscience (though perhaps sociopaths do not have a soul - I don't think there any evidence either way).
Most people however do have a conscience - even those who are generally evil. Perhaps their conscience is very weak, or other concerns (e.g. greed or a desire for revenge) outweigh their guilt. I suppose many find a way to rationalize their evil deeds, especially if they fall to it over time.
Lindsey McDonald is a great example, since he does have a crisis of conscience. In Angel 1x21 "Blind Date" he betrays Wolfram & Heart to save some children ... but is then lured back by a promotion.
Lilah Morgan actually explains explains her motivation to Angel rather bluntly:
It's a survival thing. I made a lot of devil's bargains and I stuck
to them. As a result, I live somewhat dangerously, and quite
comfortably. My mother, who no longer recognizes me, has the best room
at the clinic. I get up every morning, put on my game face, and do
what I have to...We've spent so much time and money on you, you're so
pivotal to the coming cataclysm, that I sometimes forget how dense you
can be. The game face, the one I worked so hard to get, I became that
years ago. Just like you've become simpering and good from yours.
You're the new poster boy for human? Thank you very much, I don't want
it...You think you can awaken some buried spark of decency in me? Is
that the way you "help your helpless"? I'm not helpless. I'm glad you
came along, because I was sitting here -- "what's it all about?" --
and now I know. It is all about making the rest of your eternal life
(Interesting to note she still takes care of her mother.)
As you noted, people who become vampires invariably lose their souls. It doesn't matter if they were the kindest, gentlest person before - they no longer have a conscience. They don't feel compassion. If they murder someone, they don't feel any guilt whatsoever.
Most of the people we see before and after were not merciless killers before becoming vampires. They became so only after. Drusilla is an example of someone who was sweet and gentle before being transformed into a monster. She is immediately changed into someone who enjoys murder.
It is important to note that being soulless doesn't prevent a vampire from doing good, they just have their own reasons for doing so. Obviously, Spike is the key example.
Spike ends up joining up with our heroes simply because the Initiative's implants prevent him from killing people - and he wants to kill something, so why not kill some demons? Most importantly though, he's fallen for Buffy - something he doesn't realize himself until later.
Spike does love Buffy, just as he loved Drusilla, and goes to considerable risk for her. Most notably, he helps save Dawn from Glory. He doesn't do this out of general compassion - his reward is that he earns Buffy's gratitude. While he's working with the Scoobies it's easy to forget he lacks a conscience - after all he does so much good. (Well, Xander doesn't forget.)
Buffy is forcibly reminded of what Spike is in 6x15 "As You Were", when she finds out Spike was the one trying to sell dangerous demon eggs. The result of this is she breaks off her relationship with him because she recognizes that no matter how many good things he does, he lacks a conscience and is going to do evil things too.
She is more forcibly reminded of this in 6x19 "Seeing Red", when he tries to rape her. She tells him that he's irredeemably evil, and that's why she could never love him. This precipitates Spike's decision to get back his soul. He desperately desires her love, and finally understands that she can't love someone who lacks a conscience - so he seeks to restore his soul, which will restore his conscience.
Vampire(s) With a Soul
Restoring a vampire's soul doesn't make them good, it just restores their conscience - which enables them once again to feel guilt and remorse.
Bear in mind, those with a restored soul didn't choose to be evil. They didn't find their way to it over time. They never had to rationalize their acts or learn to suppress their guilt. They're just suddenly confronted with all the horrible things they've done.
When a vampire's soul is restored they restored to whatever moral state they were when they first became a vampire. Neither Angel (a.k.a. Liam) nor Spike (a.k.a. William) were evil men. (We don't really know about Darla, even what her name was.)
Imagine you woke up tomorrow morning and suddenly remembered murdering a bunch of children in cold blood. You would probably freak out! Now consider that Angel, Spike, and Darla would be responsible for the death of hundreds, some of which were killed in particularly vile ways.
Note that having one's soul restore doesn't undo every change - Spike remains a badass rather than a weepy poet, while Angel doesn't become a drunk.
Of course, it doesn't take Spike a century to get over it. Maybe that's down to Angel / Liam being naturally more whiny. ;-) Or it could be down to Angelus having been so vicious, emotionally tormenting his victims rather than simply murdering them.
Out of universe, I think part of the purpose of the having vampires be soulless is to make them unambiguously evil, so they can be murdered without any moral issues. I think this is also the reason for their faces changing to be monstrous. I don't recall if that has been stated explicitly though - anyone have a reference?
As an aside, I do wonder how this relates to demons. I'm not sure whether it's ever stated if they have a soul in the same way humans do. Many are clearly evil monsters, but some are relatively benign, while a few are truly good. In Buffy there is Clem, while in Angel there are lots of examples of demons that aren't merciless killers, with Lorne being the most notable example.