How would the system design fit in a very tight SCRUM organization?

First, I would evaluate if having a formal system design process is correct for your organization. I don't know what level you are or the company you work at, so this is hard to tell. Here's some thoughts in a vacuum:

• If you work at a small startup or a small, leaner org in general (<5 engineers), it's possible you don't need to do this. You don't want to be adding process at this stage.
• Another reason to not do system design is if you're already shipping high-quality systems. Small companies and teams are often able to do this as the tightly-knit team is naturally able to stay in sync and small orgs tend to not have much tech debt.

If you are going to formally bake in system design into your agile scrum system, just turn it into a task and assign points to it. That's what I effectively did when planning timelines for big projects at Meta: I would carve out 1-2 weeks just for the system design and planning. The failure mode (as you alluded to) is when you add on this extra thing but the existing timeline stays the same. Now you have too much on your plate as doing both the system design and execution is too much.