Video sync generator IC
What is difference between sync generator and separator?

Big difference. The generator creates all the extra pulse information while the separator strips off the extra information to extract only the H sync pulses.

Can I use MAX9568 circuit in place of SAA1043 circuit? Would that work?

NO
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The SAA1032 Sync Gen is mainly for the VCR standard input &amp; Sync while the  MAX9568 Sync Separator is only for Composite Video input. (Yellow phono plug)
Consumer VCR IC technology is getting or is already obsolete.   So keep looking.
These are common output signals.


Can I replace this card with a Microcontroller based sync generator card? Is this doable?

Maybe
First you must master your understanding of how it works.