Should I enter income from a 1099-NEC manually on the Sch C since it's not carrying it over automatically?

In Pro Series you link it to the sch C from the NEC input screen.

Just enter it directly on the schedule C and delete the 1099 entry.  1099NEC forms inputted into software don't get transmitted to the IRS anyway.

Your client's customers might never need to send a 1099-NEC at all. That's not your source for their revenue. Their business records should list all of their business income, and this informational form is just one of those customers who paid them and happened to hit a reportable limit. Your client might make $3million, and if no one customer pays $600 or more, there would be no 1099-NEC ever issued by any of them. What you should do is confirm that their business records include all of their business revenue and do not add 1099-NEC to the number they gave you as their total, because 1099-NEC reporting would be a subset of the bigger number already provided to you.

Which is why you don't need to enter 1099-NEC when it is Schedule C. Enter the business info and confirm it's at least reasonable against the 1099-NEC amounts. Otherwise, your client might be underreporting gross, or only tracking Net, and you need to know these issues to resolve them with that taxpayer.