It would be a remarkably narrow law that made it illegal to do something client-side but not server-side. AFAIK it's usually about what data you collect and how you use it, not precise details about how it was collected and stored along the way.
Server-side allows businesses to defer the data transfer until it's known whether specific consent is granted or revoked. It also allows you to more easily keep a record of the data shared with other parties in the event that a user withdraws prior consent or invokes a right to be forgotten. You then have the ability to tell your partners to also delete those data points.