Yes, the letter from the LA should have stated why you were refused admission. Simply stating oversubscription is not enough. However, you won't win an appeal on that basis.
Oversubscription and infant class sizes are not the same thing. Oversubscription means they have admitted up to the published admission number (PAN) and some people didn't get places. Infant class sizes means that admitting any more children will leave the school with classes of more than 30 children with a single teacher in Reception, Y1 or Y2.
The fact that 46 places have been offered does not mean there are 14 places free. If there were any places free they would have been offered to the people at the top of the waiting list. My guess would be that PAN is 45 and that they have 3 classes covering Y1 and Y2. That would mean any appeal would be infant class size in that, although they would not be over the limit in Reception, they would be in Y1/2. Unfortunately that means you are only likely to win if you can show that a mistake has been made. For example, did they place your daughter in the correct admission category? Does the home to school distance they have used look right?
If PAN is 45 (you can check this in the LA's primary school admissions booklet) they have admitted one over PAN. It would be worth finding out the reason for that just in case it reveals something that would help you. However, the likelihood is that the extra child was admitted due to a mistake, in which case it will make no difference to your appeal.
Stating in a telephone conversation that you intend to appeal may not start the process. You need to submit your appeal form.
It is a shame you have rejected the offered school and only named one other school that you would be happy with. That may look to the appeal panel like you are one of those parents who think they can bully their way into their preferred school by rejecting all other offers. That could damage your appeal as the panel may be less likely to give you the benefit of any doubt. Any new offer from the LA is likely to be at an unpopular school even further away from your home than the school that was already offered.
By the way, the LA has already discharged its legal responsibilities towards you by making an offer. They are not obliged to come up with another. Many do not. The fact your LA will find another place for you is good but they won't go on doing so indefinitely. If you go on rejecting schools there will come a point where the LA will have to say it can't help you any further.
On the information you have posted I don't actually see a case for appeal. That doesn't mean you shouldn't try. You can talk about any features the preferred school has that are missing from the allocated school and how they will be of particular benefit to your daughter. It may be that evidence of a mistake will emerge during the hearing. It does happen sometimes. But I'm afraid you need to be realistic about your chances of success at appeal and start thinking seriously about plan B.