There are no pervasive reports here of bad batteries doing damage to the mainboard specifically. So my assessment is that mainboard damage from bad batteries is extremely RARE -- Unless it leaked.
But if the battery was swollen, it can push up on the trackpad, and can cause erratic operation and possibly additional damage to internal components.
These batteries are generally glued in place with industrial-strength glues. The general repair procedures for battery-ONLY repair requires that EVERY component be removed from the chassis while the hazardous solvent is applied to soften the glue.
That means if you did a home or third-party repair, there is more than a trivial risk that something got damaged while removing or re-installing, or did not get replaced properly during the repair.
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Apple does not replace batteries (in the field) at all. They swap the battery/trackpad/keyboard/top-plate assembly out as an assembly, and send that module to a factory for refurbishment.