Why? Productivity continues to increase, regardless of population increasing. It's like saying we can do more with less everywhere, except when it comes to taking care of people and the elderly.
Taking care of the elderly is a labour intensive business. Automation doesn't clean an old person's glasses (real life example), make their bed or cook their meals.
Automation doesn't need to do any of those things (even though it eventually could). It could simply make other industries so cheap and efficient that more labor flows into elder care.
Specifically, if there are socialized "assisted living" facilities, or even in-the-community caregiver visitation, then the amount of caregiving work-hours per day per elderly person is X. If the only possible arrangement is getting a live-in personal care-giver, then that figures is 2X or 3X (or some factor, I don't have the figures); and most people probably just don't get care.