Would there be any issues with setting up an old iPhone as a ‘burner/landline’ type phone?

You would need to register and pay for a 2nd line with your carrier.

And you can share Apple IDs between multiple devices. You should only do this if the same person will use both phones.

Have you considered getting a Google Voice number for your purpose? They’re free, and they can forward calls to one or more phones, hiding the numbers forwarded to.

+1 on Mr. Finch’s Google Voice (GV) suggestion

Given the rampant spamming and phishing; we generally only place our “GV” numbers in circulation.

Our “real” mobile numbers have become the modern equivalent of the earlier — landline-only — era’s “Unlisted Numbers”

(i.e. shared only with those explicitly trusted or in those cases where “RELIABLE completion of the incoming call” is THE priority.)

I’m of a similar mindset.

I absolutely don’t have Google’s app installed on any of my devices …

… rather we use GV to “call forward” to mobile numbers.