Boyfriend flirts with female friend

Really not sure what's going on in my situation to be honest. I really need advice so I can make a judgement on whether this is right or not but...my bf has been friends with this girl for the past two years we've been together and even before then. She let him live at her house ( this was before I knew him) when she moved away for a few months so I thought that was already strange enough. He constantly tells me that they are not into each other. Nothing more than friends. Recently saw his messages between them on accident and...he messaged her on his birthday saying he wants her for his birthday :( The rest of their conversations were very flirty. He made comments saying she's flawless (these messages were from other conversations) and she would send heart emojis to him. I confronted him about everything. He claims that's just the way they talk to each other. I asked him what he meant by him saying I want you for my birthday to her and he claims that he meant it in a friendly way...she moved away permanently so he says he meant it as in he hasn't seen her in a while and would want to see her on his birthday. I guess that's fine...but still doesn't make it ok for him to be talking to another girl like that, right? I questioned him about why he was commenting on her being flawless and being flirty...he told me that he only means it in a friendly way. She was kind to him and gave him a home when he didn't have anywhere to stay and he says that he feels like the only way he can give back to her is by being her emotional support for the fact that she doesn't have anyone where she moved too. She's a single mom...supposedly her ex husband was horrible to her I don't know the details, but my boyfriend claims that he's just being that emotional support like a brother. I mean...do you think this all adds up? Based on everything he's telling me it's just a platonic relationship? The way I see it is she's into him..she likes that he gives her that attention and I think he's in the wrong for giving another girl that type of satisfaction when he has me his girlfriend. She's been a problem in the relationship ever since we started dating. So after what I saw in the messages this time around I had enough. It's been two years and I couldn't handle anymore, it was making our relationship very unhealthy but he didn't see it that way for the longest time because he claims he's not into her like that. I had to give him an ultimatum. He said he doesn't want to break up over this and that he loves me blah blah. I'm not the type of girl to say that he can't talk to someone but in this case I felt that it was appropriate. I told him that he has to stop talking to her I want him to tell her and i have to be there when he does it. He didn't make a big deal about it at all just said that's fine and sent her a message saying if you don't hear from me it's because our relationship is clouding things with my girlfriend. I still don't know what to think. He never cheated but I think the flirting is cheating enough.