Jim,
Only a handful of churches have records for the period you need. Possibly 3 in Co Antrim have records for someone born in 1700. Most congregations either weren’t keeping any records then or if they did they have been lost. Those with records are Antrim town Church of Ireland (starting 1700), Antrim town 1st Presbyterian 1675, Lisburn Presbyterian 1689. The records are not on-line. There’s a copy in PRONI in Belfast but a personal visit is required to view them. If you are unable to go yourself, you could employ a researcher. Researchers in the PRONI area: http://sgni.net
Belfast was very small in 1700 (the capital of the province at that time was still Carrickfergus). Belfast in 1700 wasn’t much more than a bridge over the river Lagan. Population of the town in 1700 was 2500, so the family may not be from Belfast.
Researching in Ireland in the 1700s is very hard going due to the general lack of records. If you don’t know where they lived it’s a needle in a haystack. Ideally you need to know the person’s exact denomination and the townland or parish they lived in to have any chance of finding them, and even then there may not be any records for that location.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).
You ask where the family may have sailed from at that time. All sorts of ports were used. Belfast, Newry, Carrickfergus, Larne, Portrush & Londonderry all had sailings during the 1700s.
Though there were earlier migrants from Ulster to north America, the main wave of migration started in 1718. Primarily Presbyterians whose ancestors had arrived from Scotland in the 1600s but were dis-satisfied with life in Ireland. Having already migrated once they were open to moving on again. Today they are known in north America as the Scotch-Irish and in Ireland as Ulster-Scots. Some background here:
https://www.ancestryireland.com/wp-content/uploads/2017/08/The_1718_Migration.pdf
Plenty of records on their subsequent lives in north America. Very little known about their lives in Ireland.
