The problem is this. Lillies effect around 6 CDs I think and whenever one of those CDs is used it resets the Lilies. Now on average if you are spending all your GCDs spamming Cure or Cure 2 you are likely to get 4 to 5 Lillies optimistically. A number of those CDs are on 1 minute cooldowns. This means, unless your not actively using CDs when they are available you will be using Lillies before you reach three, probably most of the time only with 1 Lilly. Lillies cd reduction is exponential. So having only 1 Lilly means only a 4% cd reduction, which for most of the CDs it affects is basically around 2 second.
Fundamentally you will probably be saving a total of maybe 10-15 seconds off all 6 CDs in a minute and with little control over which ones get the most CD unless you actively don't use CDs to save the lillies for a particular CD. To me seems a little counter productive and devaluing of the mechanic cause it means restricting CD use.
Note, that is if you spam nothing but cure and cure 2 constantly.
The issue is that from a player's perspective this seems like an obvious issue. Something that seems like the devs should have spotted. The only time you have designs that seem strange like this but make sense in context is if the gameplay has been altered for the design to fit. However if that was the case the other heals haven't been altered really to compensate for such a change which leads to just more balance issues.
Honestly I don't know why they don't seem to be picking up on this but they seem to have been having trouble understanding the problem since AST was introduced. Fundamentally the design of their Jobs doesn't seem to match with the healer gameplay design of FF14. Its building a street car and then making it do offroad driving. There are going to be issues. The design doesn't match the environment it functions in.