Most obviously, because she wouldn't have had the option of using the house as security for a loan to pay for a carer, or for other things to improve her quality of life — physical adaptations to the house to accommodate her growing disabilities; funds to pay to move into the residential home of her choice at a time of her choice, pending the proceessing and approval of a Fair Deal application, rather than having to take whatever place is available. As it happens, she has done both of these things.
This is her house, not mine (or my siblings'). It should be used, first and foremost, in whatever way is convenient or appropriate to maximise her welfare and her quality of life. Any inheritance for me or my siblings is strictly limited to whatever remains after those priorities have been respected. The notion of giving away the house while in fact she still needs it (evidenced by the creation of a lifetime right of residence) creates a situation in which the ownership of the house is divorced from the identity of the person whose benefit the house should primarily serve. This can't improve my mother's welfare or quality of life and could foreseeably disimprove it. Risking that so as to protect my inheritance prospects (from what would be a strictly finite risk, given how Fair Deal operates) is a trade-off I would be very uncomfortable about benefitting from. Hence I would strongly urge her not to do this.
On a separate issue, I think your calculation in post #3 of the tax and other implications of the arrangement assumes that the recipient of the house is going to move into it and live there with the parents. The OP doesn't suggest that this is going to be the case (and, in my own family, this would not have been the case).
I say this because, if the owner of the house doesn't live there it will, of course, not be their principal private residence, so any capital gain accruing will eventually be brought to charge. Your calcuation of the financial impact doesn't include any allowance for this.
(And, while it's not a financial consideration, if the owner of the house does live there it creates an emotionally complicated situation, in which there would very llkely be pressure/expectations on them to take on the role of the carer that one or both parents might in time require, so as to avoid or defer the need for them to move to residential care. Mum and Dad may be sanguine about the prospect of moving to a nursing home when the prospect is ten years away; when it's upon them their attitude could easily be different.)
And one other point; your calculation assumes that the parents will live for at least three years after moving into residential care. Not to be gloomy or anything but, statistically, if they lived that long in residential care they would be beating the odds. The OP's parents are in their mid-60s and already have health complications. On the figures, €101, 250 is the maximum that the arrangement could save, but the likelihood is that the actual saving will be less than that.