Is it because REITS are a reasonably safe-ish asset but start...

Is it because REITS are a reasonably safe-ish asset but start getting replaced by a safer asset e.g bonds due to rising yields.

I'd have to start by saying that I'm not sure that the reduction in the SP of ARF is wholly due to the change in bond yields. There's probably a range of issues like there usually is with the analysis of any stock.
But your understanding is about right. REIT's are compared a bit to bonds because their dividend is generally the only income with little profit growth involved - sort of like a bond.
Therefore, when Government bond yields rise, investors expect REIT yields to rise in the same way and the way to achieve that is to force the SP down so that the relatively fixed dividend then represents a higher return on the investment.
But there's more to this investment than just that. My only reason for reference to bond yields was because the discussion was about how low the price here might go, and I suggested that when bond yields stop rising then the price may be around its low mark.
You're right about their growth. I owned this stock for many years and started to sell out about a year ago when I first reached the conclusion that interest rates would start rising.
Throughout that time, I received a 5% dividend and a 4 - 5% improvement in their result and subsequent share price giving me what I would consider as, roughly, a 10% per year return on my investment.
If you don't own the stock but are considering buying in then it would be excellent if you could get the low price before it starts rebounding although rebounding in any meaningful way may not occur until interest rates start to drop. I have no idea when that may be.

The risks associated with this company at the moment are minimal:

• they have some debt so rising interest rates may impact although they appear to have hedged enough to keep that effect to a minimum.
• if we have a recession than sometimes these Child Care Centres can bankrupt themselves and leave a hole in the rental stream for a while

Against that the Government seems to understand the need for Child Care centres so that more mothers can get to work and reduce the employee shortages we're experiencing.
Good luck - this is a solid company in my opinion despite the price reduction it's currently going through.
If you compare the return here with a Coles or a Woolworths remember that there are no franking credits available with REIT's.