Secondly, there is presently no real reason to suppose that any of the demonstrated limits to human cognitive function are caused by anything other damage: loss of neurogenesis, build up of amyloid and other waste products, progressive demyelination due to a variety of processes, white matter damage from microstrokes resulting from vascular dysfunction, and onward for dozens of other types of harm. Every aspect of memory and executive function is slowed and degraded. The detailed maps of connections between damage and dysfunction are in most cases sketches or unproven or unknown or exceedingly complicated and open to debate. But there is so much damage that it would be going out on a limb to suggest that a periodically repaired brain wouldn't just keep on going, losing the old memories and generating new ones, not filling up.
There are bodily systems that do "fill up" and where we can make good arguments that they have structural issues with long-term operation and would fall over even if every component part worked fine perpetually. The immune system, for example, in its continual assignment of memory cells for herpesviruses such as CMV, may be working as evolved but nonetheless gradually destroying its own effectiveness. That can be argued reasonably, versus it all being the consequences of damage.
But there's no reason to think that any of the brain's processes are in the same boat at this stage. There is every reason to think that dysfunction is all damage until proven otherwise.
Lastly, having the opportunity to be in good health and wrestling with a century of memories while deciding what to do next? That beats the alternatives. High class problem.