Lets try to offer a mutual assistance here.
You did it quickly, you directed straight question and she sort of agreed.
What I want to know is how did she act before you clearly opened the breakup cards with her? Do tell. What do you mean by cold and distant in a pattern of your relationship? Was it your gradual instinct or do you have blatant examples
Because after half a year of dating and relationship with my gf, I begin to notice the same thing.
My story:
This is 'cooling off' stage definitely, no mistake. She acts 'weird'
in the glimpse of the moments, or just normal or much more comfortable, though there is still affection. She puts her girlfriends as a high priority suddenly, and really doesn't act with same enthusiasm anymore next to me. Even for a day, certain days in a week, you can't look and not notice these things. There is a sudden increase of affection still, she claims she wants to be with me, however we are going into distance RS after next 3 months. Probably it gets to her. Its just obviously different, lesser élan when meeting me or even rarely initiates by herself, few small strange events which I won't go into detail now.
Another thing, when we had sex today it was just pure fucking. I thought this is required once in a while, variety, but this is the first time she wasn't emotional as usual during sex, not even for a moment. Actually she was into it a lot at first, but it lasted very long(trying new positions and things), which caused me to delay my orgasm really long, and at that point she was just jerking me off(I actually had to tell her 'I'm not finnished yet'), not having too much blowjob from her as a finnish(she does have strong gag reflex), and after everything she was very tired when we watched TV show, but that also seemed cold, as she completely avoided cuddling after sex immediately. She doesn't talk about spending time tomorrow again even though she made no other plan, and lack of enthusiasm can be bothersome. This is also the first time I got very quick goodbye and kiss when walking her home. without her usually be the first to say she loves me after we spent night together and had sex. Tonight I had to say it first.
Then I noticed she left her FB page logged in behind her. I didn't want to search inside, and there is nothing really to look at, which she confirmed "there's nothing anyway

", as I wrote her it was open. I am a sceptical and paranoid person, but at this point I just don't think its cheating.
And like yours, my girlfriend has modus operandi of being quiet about problems, shy, not wanting anything that can lead to drama, and usually a cheerful person.
If you ask about any changes it could lead you to make problem out of it(after plenty of months person can have other personal worries besides relationship), or you can dig out real existing problems like OP did.
Now,
I thought this topic, instead of opening new ones for the same thing, could serve as a help to OP, and people with similar situations - about behaviour after a while in relationships, when you have no proof of anything, or can't be sure if theres anything deeper to it, but things just DO look different. Should you talk to them about it, and pull information out, as they obviously always say "all is fine, I'm fine, its nothing, blah blah".
What she answered when I asked her subtly just "whats up" over this: "its just because I'm acting normal at the moment, not being hyperactive with you?" or something along those lines. I said: "no, no, no, you just seem tired".
Someone should write about this, its a serious detail in context of dating and relationships, and it can't be sunflowers all the time, but I don't understand why people don't strengthen their bonds through inevitable change which is a fact in all aspects of life, they rather identify with problems and loose, resulting in present situation, which is: "we accepted that romantic bonds are nothing but momentary fun waste of time, nothing else, no need to work out like for any other life challenge". One has to THINK about this beforehand, and not take "I'm willing to get into serious relationship" as a joke. At that moment you don't think "lets just give it a try". Awareness that it will get nasty, it won't be easy all the time has to exist.
Of course you won't be able to ignore her or just let it go, its already sealed. But general mentality is just "dump" mentality. Something you WANT doesn't appear too easy, and "you should just drop it". If this was about other life decisions like career, no one would get anywhere with such way of thinking.
Thanks for reading my rambling and personal stuff, let's make something constructive out of this topic for everyone.