"Three is not very reliable network as they have not full coverage."
Which is irrelevant if the phone is being used for roaming within Europe, including Ireland. When roaming your phone will prefer the local 3 service, if it exists, otherwise it will use any available network. When used in the UK there are 3 network black spots but there are black spots with all networks and sod's law says that no matter which provider you choose you won't find coverage when you are desperate.
In the (long, broad) valley where I live in the UK 3 is the only consistently available network. If I travel out of the area there are black spots in other valleys but usually limited to a radius of a mile or so.
It's not true to say that they are "not very reliable". The network is reliable, it just has patchy coverage in some areas within the UK.
In Italy I get the best network coverage with 3 and since all data is included in my plan it's remarkably cheap. Much cheaper than getting a TravelTalk SIM or a local SIM.