Will I have to always enter the UK even on the second trip first before I can make my way through to Dublin?
You have an entry clearance issued under the British-Irish Visa Scheme (BIVS) and have entered the UK previously and want to know if this counts as meeting the requirements towards a separate trip to (or through) the Republic of Ireland.
No, once you leave the Common Travel Area, your BIVS 'resets' in the sense that the original requirements must be met again. Your first entry must be in the country that issued the visa.
The standard 'go to' page for BIVS is British-Irish Visa Scheme, and this scheme is not explicitly addressed in the rules themselves. It is a policy created by Ministerial Authorisation by the previous government, and subject to withdrawal by either side. Also note that information submitted to either side is made available to the DHS.