Will airlines allow passengers to bring outside food on the plane?

Yes, you can bring food on the plane.  You can even bring food through security as long as it doesn't violate the liquid/gel rules - I do it all the time.  As far as I know, all airlines allow you to briefly remove your mask to eat and drink, but you should do so very briefly. 

Thin gs change all the time, but I have flown on two Asian airlines that didn't allow passengers to eat their own food on the plane: you had to buy from the plane's own menu.

Of course. I flew in January, and took some snacks on. You can bring solid food through security- just not liquids and gels. Then you can purchase things in the airport and take them on. You need to be considerate and take your mask down just long enough to take a bite or a swallow. 

What do you mean, will they? They allow it now.

Yes, you can still do that.

You can't bring anything like that through security, but once you're through, you can buy something in the terminal and eat it on the plane.

At least, you could the last time I flew.  Which was admittedly a month before the pandemic.

Yes you can
I routinely bring food from home for long international flights

I do it all the time ( or did up to the end of 2019, haven't flown since ).  
I routinely get a coffee, maybe a muffin and some other snacks to eat while in the air.