Is it allowed to use a macro to keep a button held down while flying?

Using a macro will 100% get you banned.

Don't do that, brother, you will get banned

Why would anyone even do that? Are people really that lazy or stupid?

I wouldn't recommend it. The Positive Play Charter states:

Don’t use or distribute unauthorized hardware, accessories or software programs and tools such as aimbots or triggerbots, or otherwise use exploits, cheats, undocumented features, design errors, or bugs.

My understanding of that is that doing what you're asking about would risk a ban, so it's better to play it safe. 

Here's the Positive Play Charter if you wanted to double check it:

 https://www.ea.com/commitments/positive-play/charter#treat-others-with-respect 

AcidryzeN​ there is no jet auto throttle unlike every other battlefield game. so holding down a button replicates that behavior. that's all. im not "lazy" i just don't want to retrain 10 years of muscle memory. Im pretty sure you wouldn't like someone messing with your muscle memory too.

Idk why not. PC users utilize ADAD/kneel/stand/kneel/stand while shooting keyboard macros like a boss. 

As a console player this is yet another reason to never utilize crossplay.