The reader should review the (short) question/answer
$\quad$ Supercharging the axiom schema of replacement in ZFC?
before proceeding here.
Drop both
$\quad$ 6. Axiom schema of replacement
and
$\quad$ 7. Axiom of infinity
from $\text{ZFC}$, giving $\text{ZFC} \setminus \{(6), (7)\}$
Introduce the new schema,
$\quad 6_{sc}$ Axiom schema of sc-replacement
giving a new framework $[\text{ZFC} \setminus \{(6), (7)\}]+(6_{sc})$.
Consider the following argument:
Let ${\displaystyle S(x)}$ abbreviate ${\displaystyle x\cup \{x\}}$ where ${\displaystyle x}$ is some set. So we have a mapping
$\tag 1 \psi: x \mapsto S(x)$
defined on all sets.
Let $A = \{\emptyset\}$ be the primitive singleton set. Applying axiom $(6_{sc})$ there is a set $B$ satisfying
$\tag 2 A \subset B$ $\tag 3 \psi[B] = \{\psi(x): x\in B\} \subset B$
The set $B$ is not finite.
Are we back in 'ZFC land'?
Notwithstanding the assistance given by Noah Schweber (background post) and Andrés E. Caicedo, a detailed proof showing how
$\quad [\text{ZFC} \setminus \{(6), (7)\}]+(6_{sc}) \text{ implies } (6)$
would certainly wrap this up. I would attempt it myself but it would be an 'all thumbs' mess!