Should I Ask for Promotion?

In my experience, you don't get what you don't ask for. In other words, your work/performance rarely "speaks" for itself. If you believe your work is representative of a promotion, then you should absolutely talk to your boss about it. Even if they say "no", you can follow up with "what do I need to get there?" and demonstrate a willingness to grow and level up in the organization. That can kickstart a process where your boss is actively helping you get a promotion. (Assuming they're a good boss...)

For now, I recommend compiling a list of achievements and work that you've done that illustrates your growth. When you have this conversation, your job is so sell yourself as worthy of a promotion. You can't just say "I'm working more". That won't fly. But you can say "In the last 6 months, I've accomplished X, Y, and Z. I proactively identified problems M and N, and recruited Joe and Sally to develop a solution, then implemented it within six weeks. This solution saved/made the company $A, and increased productivity by B%." Numbers and data generally speak more than words.

Another piece of advice that a former boss told me is "don't ask, tell". I'm not sure I'm 100% on board with this, but the idea is that you go in with something like "I've done all of these things (see above), and it's time for me to move up to the next level and get a promotion to Senior Whatever." This shows confidence and assertiveness, which is generally a good thing the more senior you get. I'm not 100% convinced this works, but it has worked with me for the last few bosses I've had, so... YMMV.

Good luck.

It depends?

I.e. when I worked at a fairly large multinational (around 10k headcount), the only time I got promoted was when I explicitly started asking along the lines: "I like the sound of 'senior engineer', how do I get there?"

But there was a process. Maybe you don't have a process.

W.r.t. 'having your performance speak' I would suggest looking at it as 'building your case' rather than 'I am working a lot, right?'

Doing lots of your current job != exhibiting you will succeed at the sorts of tasks if you were promoted, which is why if you are great at a certain job you can get stuck doing it, especially if you're a bit understaffed which I guess you are if you're doing a ton of extra work to keep up.

Can you pick up / start doing work that'd make it clearer you could be a leader (assuming that's what the next step looks like) without dropping the ball?

If you can actually repeatedly and successfully create and grow new products and services, I would start your own company.

Like a-saleh says, about 'building your case', you want to position yourself such that it's undeniable you are producing more value with your time. Point out that there has plainly been a shift in the value of your time for the organization.

You can probably ask for anything you want. I would doubt, however, that anyone would give you anything unless they a have reason to do so.

Like an outside offer.

Why would I, as a company, promote you and give you more money when no one else wants you? I have no incentive to do so.

I’m sure wouldn’t do it out of the good of my heart. That doesn’t exist. People may try to make you think like such thing exists and you might get a promotion from time to time but that is a fraction of what you would get if someone else was interested in you.

Yes, make people say no.

That said, switching jobs is generally the best/easiest way to move up.