I'll assume, by Chrome you mean desktop Chrome. This is irrelevant to my comment. I was specifically talking about realtime push notifications via mobile devices. Also remember that your preference is just your preference, many people would want to be able to make sure their messages get seen in realtime. And Facebook probably wants that for their users too. Again, I'm not saying this is a good thing. I'm just saying it's not an illogical decision. You are free to complain but it would be stupid for Facebook to keep going this way just so they can support the legacy technology.
Chrome on Android does support real-time push notifications. I know because I was notified of a friend's Facebook post I was mentioned in within 5 seconds of my friend posting it. I don't have Facebook installed on my phone, and he posted it from his phone while my phone was face up on the table between us and we both where surprised at how fast I was notified.
Please read my other comments on this thread. Basically just because Android supports it doesn't mean it's all good since iOS doesn't. Also the "open web" you think you have on Chrome is not really that open as you think if you think about the entire mobile ecosystem landscape.
I keep saying this in every comment I made on this thread, but I'm talking about the entire mobile ecosystem. Who cares if Android Chrome supports it, iOS doesn't. Which means it's not a 100% coverage. Just by letting this happen you are providing a low quality service to not just the iOS users but also the Android users, since there will be an assumption that the recipient doesn't always get the message in realtime. If you want to send that kind of messages, there is always email. Messaging apps are built on assumptions that people know the other person receives push immediately.
1) you can never assume a recipient ever has their phone on, with them and in a non-silent mode
2) users can disable push notifications for individual apps anyway
For some reason I feel like I have become some sort of FB defender, but I am not. Just to be clear, I am not a fan of FB. I even completely got rid of my FB account a year ago because I got sick of fake interactions on the site and have never looked back. I am happy with communicating with close people just via text and other special purpose communication apps.
That said, the reasons you mention are pretty edge cases. Instead of just hating on it, just stop and think about it for a moment without prejudice. When you email someone you don't expect them to read it immediately, but when you send a text you at least expect them to know they received something, no? (If you say no to this, I really have nothing else to say anymore, you would be just denying the reality) That's what Facebook wants to be. They're competing with SMS, not email.