It was very common to wind up an estate without obtaining probate. It happened all the time, and wasn’t illegal. I am not sure about the Republic of Ireland today but in Northern Ireland and the rest of the UK it is still possible wind up an estate without probate, where the estate is of very low value or all the main assets are in joint names. But in the 1800s it was very common. The main assets were typically the farm, and moveable items such as the cattle, machinery, seed and the family possessions. A little cash in the bank perhaps. There often wasn’t anything else. Probate tended to be essential if there were insurance policies or large investments. Where that wasn’t the case, the relatives simply distributed the bequests in accordance with the will, or in the absence of a will according to custom. Banks often used to hand over small deposits to the widow or son without sight of a probate document.
Absence of probate had little impact on property. Land registration has only been compulsory in Ireland in very recent years. In the 1800s it wasn’t necessary to register a transfer of ownership, and in any event all that was usually being passed on in most families was the unexpired portion of a lease, as very few people actually owned their land anyway. But even where you did own the land, the person who had inherited it under the will simply took possession. Should they later decide to sell, they would gather together such evidence as they had of their entitlement to possession eg any previous deeds and the will. They might add an affidavit confirming they had acquired it through inheritance, which is why they were now the vendor, and that generally would satisfy the purchaser. (There was little other evidence they could produce anyway). That system worked fine.
You ask about undertakers bills, and how does the undertaker know he is going to be paid. The short answer is he doesn’t. (Even where there’s a will and probate he’s not necessarily going to be paid if there are insufficient assets). In rural Ireland business transactions have always been done on trust. That is still very much the case today. People are usually happy to bill you after the service you asked for has been provided, and not just for funerals. They sometimes make a judgment based on knowing you and your family. There’s no mechanism to guarantee payment though. It would appear from what you say that non payment of undertakers bills may be an issue in the US, but it’s not in Ireland. The undertaker will normally be paid. It would be a serious breach of trust and faith if that didn’t happen. No need for legally binding agreements. People know what is expected.
I do research mostly in Northern Ireland and whenever I encounter a death, I routinely check the PRONI wills site to see if probate was obtained for the deceased’s estate. Mostly it wasn’t. I’d estimate that, until the 1930s or 1940s, probate was obtained in no more than about 5% of deaths. The rest were wound up in the manner I have described above. More recently, most deaths would lead to probate being obtained, but your enquiry was about the 1800s and early 1900s, and so as I say, no probate and no obituary was the norm.
In a small rural community it isn’t necessary to advertise a death. Everyone in the area knows anyway by word of mouth, and anyone with an unpaid debt will call and see the family or executor, at an appropriate time after the funeral. (The norm in Ireland is to bury a person 3 days after their death).