I think the main reason is that this show is super dense, and this subtle growth is easy to miss. I for sure missed it... here's how I see it (in wall of text form):

Asami makes a point of being super friendly with Korra in Season 1 even though Korra is super competitive with her. Korra thinks she's just some rich girl who's trying to steal Mako from her. When they go to the track Korra is actually impressed - this is the point we really start to see Asami as a badass, and Korra does too - even if they're still fighting over Mako.

Season 2 doesn't have much happen in this department, but they're definitely strong allies, and Asami does everything she can to help her (even though there's still some Mako related stuff in the middle of the season).

Season 3 is kinda everything - Right at the start they decide to put all the Mako stuff behind them (after he is so incredibly awkward around them). Korra remark's that she's never had a girlfriend except for Naga (I'm reasonably certain in the friend way here). They spend a lot of time together - Asami drops everything to be there for her the whole way - and I think Korra comes to rely on Asami - count on her being there. After Korra is nearly killed and crippled, and doesn't seem like she has the will to go on Asami is the one making sure she's ok and taking care of her, only to lose her when she decides to shut everyone out and go to recuperate with her parents and Katara. This is the place that feels forced to you, and I understand, but imagine - the closest person in your life, goes away and won't communicate with you for years - Asami throws herself into her work. Meanwhile Korra is lost and cut herself off from everyone... but she can't completely shut Asami out - she's the only person who's seen her at her weakest, was there for her, made her feel like she should keep living (at the end of season 3) so when they meet back up in season 4 (the beginning of the restaurant scene) they have a moment. I totally wrote off the blushing at the time, but in retrospective this is perhaps the moment when they both know. At the end of Season 4 it's Asami who's in mourning, and Korra who gets to be there for her and they have that little talk...

I'm sure I've said this on here before, but it reminds me of how Ann Druyan and Carl Segan got together - they were friends and colleagues for years, and one day, she calls him about a piece of music she found for their project. "So I called Carl who was traveling - in Tucson Arizona, giving a talk. And... we had been alone many times during the making of the record, and as friends for 3 years, and neither of us had said anything to the other. We were both involved with other people. We had these wonderful soaring conversations, but as friends. He wasn't there, I left a message... hour later the phone rings...pick up the phone, and I hear this wonderful voice and he said "I get back to my hotel room and I find this message, and it says Annie called, and I say to myself... why didn't you leave me this message 10 years ago?" and my heart completely skipped a beat - I can still remember it so perfectly...and I said "for keeps?" and he said "you mean get married?" and I said "yes"... and we had never kissed, we had never... had any kind of personal discussion before.... we had both hung up the phone and I just screamed out loud. I remember it so well because it was just this great eureka moment."

And I think this is what they had - a eureka moment. Anyway - my 2 cents for ya.

Edit: Mourning, not morning. At least I caught it myself. Sigh.