People complain about this permissions issue a lot, but it's not hard to solve. What I do is have a sheet which I name "Splash sheet" into which I paste the URLs of the documents I wish to link. To its right is a column headed "permit to connect" which contains IMPORTRANGE formulas importing a single cell from each sheet -- usually a cell containing a confirmation code, number or document name -- on a sheet also named "Splash Sheet." For example,
=IF(B3="enter URL",,CONCATENATE(IMPORTRANGE(B3,"Splash sheet!A1")," ",IMPORTRANGE(B3,"Splash sheet!B1")))
So, when you first connect a spreadsheet via its URL, you get those messages telling you you need to connect, you click the Permit Access, the confirmation code/number/document name appears in the second column, and voilá, your sheets are connected forevermore! Now all your other IMPORTRANGEs referencing that URL will work, and you can use IMPORTRANGE formulas that reference the URL-containing cells on the "splash sheet."

As for the OP's original question, I came here seeking an answer to the same problem, and after more research have realized that we are attempting the impossible here. No way to do this an ARRAYFORMULA. No way around writing formulas that reference every single cell a document's URL may go into.
Problem is you can't make arrays of arrays in spreadsheets; that would involve multiple dimensions, and the medium is inherently two-dimensional. This is what people use databases for.