Anecdotally, in the tech industry you usually will need to change companies to move up. Certainly not true everywhere, but it's been a "feature" of the industry for a decade or more.
That does seem to be the case here.
Learn the things that no one else is willing to do or easily has the capability to do and ask for a raise or move in position. Be prepared to move on. They will give you what you want.
There really aren't tasks of that nature where I work.
We also have a policy of ensuring that we have no single point of failure when it comes to technical or institutional knowledge. Our tech work community is really good at ensuring that we keep others in the loop and document everything.
Non-management positions are subject to a union collective agreement. The money is good mind you but there are no raises outside the pay grids.
You are not popular enough at your work place or at least not with the right people. Change jobs and try again.
Advancing through technical ranks does not qualify one for management. It qualifies them for more technical stuff.
Are you taking leadership responsibilities? Leading your team, mentoring others, representing your team in broader meetings with other teams, helping with planning, etc. Doing that stuff probably means you stop doing technical stuff.
Yes. I'm currently a team lead, train or tutor more junior techies and have had great performance reviews.
I have also taken the company recommended leadership courses.
I have seen this in many orgs. It is sometimes hard for others to see you in a role other than the one you are in, or sometimes they see you as too valuable in your role to let you leave it.
I agree with the others that sometimes the path for career advancement can be to move on elsewhere.
Find another org willing to support your professional development. "Thank you for the opportunity to grow here, but there is no longer a path forward here at this time. I am willing to entertain returning if that changes in the future." once you find your next role. Leave the door open to come back in the future when they can provide what you need, don't burn the bridge. Sometimes the only move is to leave.