Hi,
As you've mentioned possible half-siblings I've had a few more ideas...
So as you know, some people test DNA on one site, others on another. Therefore "fishing in all ponds" MIGHT help. (Or it might help in the future as more people test). As you say, you've already tried Ancestry & MyHeri, but there is also FTDNA "FamilyFinder" and LivingDNA where you can upload your autosomal DNA raw data or buy a new test. 23andme also do an autosomal test but you would have to buy a brand new one. (I've not tried this one yet).
If funds allow, there is also Y-DNA testing from FTDNA which would give you a Haplogroup - if you were assigned one which is very common in Ireland then it would look likely that your bio-father's father's father's etc line is from Ireland in the last few hundred years. If you discovered your Haplogroup was, for example, R-M269 this would be more tricky as it is common in men across the UK, Ireland and Western Europe. (But you can join a Project on FTDNA & the volunteers are very helpful in figuring out results I've found).
Regarding your trying to link 2 families together, could it be that your bio-father was born to a pre-marriage couple and adopted (informally/formally) as a baby? Perhaps he was brought up by distant family or an unrelated couple (in England?) If you bio-gt/grandparents then married each other and had more children you would have quite big DNA matches to the later children....
As you say your 500+ cM match is also puzzled about the link, perhaps he/she genuinely had no knowledge of your bio-father's existence for whatever reason. That is, until you did a DNA test.
Of course this theory could be completely wrong but you never know where exploring ideas may lead.