Does page verification mode affect DBCC CHECKDB behavior?
It is highly recommended to set a database to use Page Verify CHECKSUM.The Checksum is safer and its performance overhead is still small enough.
As discussed in this article.

When changing a database to use CHECKSUM it is important to know that
the option isn’t used for pages that have already been written to disk
and as such it will only be applied to new pages are written or when
existing pages are modified and re-written to the IO subsystem.
A common misconception regarding the CHECKSUM option is that it
replaces the need for consistency checking databases. This is a very
incorrect assumption; as we’ve seen the CHECKSUM error is only
reported on the read operation of an inconsistent page, if the page
isn’t read the error isn’t reported. However the biggest reason is
that the DBCC CHECKDB routine performs a lot more thorough checking at
the database level and includes checks that a CHECKSUM simply doesn’t
cover, therefore the best practice is to combine CHECKSUM page
verification and regular consistency checks.