//One of the awards is Member of the British Empire. It doesn't make a lot of sense to be awarded the membership of something that doesn't exist.//
Actually that’s not quite correct, ludders. And the difference, which may seem pedantic, is critical to the point you make. An MBE is actually one of the five ranks of “The Most Excellent Order of the British Empire". All the five ranks of the order (GBE, KBE/DBE, CBE, OBE and MBE) are similarly named and relate to the Order of Chivalry under which the awards are made. A person receiving an MBE is therefore a member of the Order (which still exists) not a member of the Empire (which doesn’t).
So the Empire no longer being in existence is really no bar to the awards being made any more than Victoria or George VI no longer being around is a bar to their eponymous medals being awarded. You might just as well suggest that the Order of the Bath is no longer appropriate as the recipients are no longer bathed as part of the award ceremony or that Knighthoods should no longer be bestowed as we don’t have men in suits of armour going round rescuing maidens in distress.
//The British Empire Games ceased in 1950. Presumably because there was no Empire for us to compete against.//
In fact it was 1974 before the word “Empire” was dropped from the title of the games. From 1954 to 1970 they were known as the “British Empire and Commonwealth Games.” Those held in 1974 and 1978 were the British Commonwealth Games and in 1982 they took the title they still hold, the Commonwealth Games. But by 1968 all bar Southern Rhodesia and a smattering of islands and outposts had gained their independence and it seemed appropriate that the games were renamed about that time as the games would have consisted only of competitors from the UK, Southern Rhodesia and a handful of outposts which may have had trouble raising a relay team.
//I note that the highest Italian gong is the Order of Merit of the Italian Republic, and not the Order of Merit of the Roman Empire.//
Presumably because when it was originated in 1951 the Roman Empire no longer existed. But I could be wrong. The UK Orders of Chivalry were originated in 1917.