We used similar lines for the warm water on a semi automated refill.
I added a temp gauge and set it for 81F.
This goes to a post carbon filter with 40lbs of AC in it and then the tank.
Drain is a simple solenoid and refill is a auto float.
Problem is that the drain is much faster than the refill.
But it's semi automated, I just drain and the tank manually, then refill, no timers etc associated.
If you want, using smaller hard plastic RO type line for drain and refill can be done much simpler and in line with the canister filters. The drain and refill is slower, there more mixing refill/flushed water, but install is much less complicated.
Solenoid for the drain only, auto float (dual for redundancy) for refill from the hot/cold water system. You can even use the cold water if there's a decent water heater in the tank and the refill rate is slow/use small % water changes.
You still have no chlorine removal however in those schematics.
The little filter in the laundry room does not look like it can do much more than filter mechanical stuff. I'd add a dual 9.75" GAC or solid block filter and run the refill slow.
With smaller hard plastic line, it's very easy to work with, trade off is slow refill/drain. If you go large drain/fill pipe, then it seems to work to run it semi automated, turn a valve to drain 50% of the tank, turn another to refill.
Very little motivation required.
Low chance of disaster also.
It can be done either way there with the set up you have.
I'm still a semi automated person, I cannot afford to flood the places no matter what.