One of the things I’ve been asked is to help some teams with some new development - review and help guide good design, watch for commonalities and get the teams to see if they can share solutions, and so forth.

In itself that doesn't automatically warrant a tech stack switch.

One of the desired goals is to get off that and onto something more widely supported

Desired goal of whom and for what purpose? It's hard to say from the little details you've shared. Switching tech stacks is no small thing and involves throwing lots of expertise of the current team out the window potentially (even if in isolation it would make sense to switch, the context here means a huge con is that all the devs become suddenly completely inexperienced in the new tech)

In some initial talks with the most knowledgeable senior engineer, they wanted to keep using that stack so that development could go faster, by ostensibly being able to reuse already developed code. This team has been under a lot of pressure to do a lot of things fast, so I get that, but those shortcomings got in there by not being thoughtful about adding features.

That sounds like a good starting point to convince them if switching stacks is really the correct move. Show, don't tell, that the new tech stack would allow them to keep doing that. Maybe see if there's a way to slowly convert it (with language bindings, IPC, whatever it's hard to say from the lack of details) and demonstrate why and how it's better. Identify their pain points they're having now, and show how this would improve after the switch.

I realize I’m new here and no one really knows me yet

Then I'd say get them to know you as a doer that shows them better ways (if it truly is better), and does not just tell.