This is obviously untrue, they can definitely do it, and there are people I’d be happy to see in those roles, but the quality of work would suffer in the short term, and we have a ton of deadlines/projects to complete.
In many places there is always something urgent and there will never be a good time to slow down. Then you end up in my first point eventually. So part of your job is to create that breathing room, push back on projects, get people trained, etc. Think long term and not just moment to moment.
edit: And if you think training people now is painful, doing so when you're actually past the breaking point will be be a lot harder and cause a lot more disruption.
I spent a lot of time training people in my previous job only for them to come back to me with basic questions about what I taught them over a year later.
What was missing was technical leadership from above: the managers didn't realize (or value) the need for the team to get familiar with the systems that they were working on, so they didn't make sure that the team learned these things well, or brought in a few more people who could guide them in addition to just me.
It’s on both of you to get at least one other person trained.
Which is better? Long term failure or short term hiccups?
In my experience, though, I see managers and therefore companies struggling exactly because they do the opposite. They are unable to delegate, to trust and to think "he or she doesn't have the right experience to do it". Sometimes this stems from pride and arrogance - you are unable to admit that someone else can do what you are doing and you should maybe focus on something else. It can go as bad as a power play: if I grow people, eventually they want more and more.
Other times it comes simply from ignorance: I have always done it this way, and I am not sure how to move forward, because nobody in the team is ready to lead.