To summarize what I wrote in several comments...
The least controversial conclusion about this election was that both Labour and the Conservatives got punished at the polls. Labor lost half their (MEP) seats, i.e. went from 20 to 10. The Conservatives lost even more, 15 seats, being left with just 4; their worst ever EU-election results. So clearly these two parties are among (if not the main) losers of this election.
On the avowed remain front: The Lib Dems are clearly a big winner in terms of swing, going from just one MEP to 16. And to a lesser extent so were the Greens from 4 to 7 and the SNP gaining one MEP for a total of 3 now. Plaid didn't make any inroads, keeping their only MEP. The new party Change UK didn't get any. In total this remain group has 27 MEPs now.
On the avowed (hard) Brexit front: The new Brexit party indeed is the largest winner as a single party with 29 MEPs; they outdo the avowed remain parties by 2 seats. But in terms of swing it gets dicy: the UKIP lost all of its 24 seats! So the net gain in this camp was only 5 seats, a much less significant momentum. If Wikipedia is correct on this issue that 14 former UKIP MEPs had already formally switched to the Brexit Party before the election, then the swing of the Brexit Party itself in the election is only 15 seats (on par with the Lib Dems).
Finally, there's the thorny issue how to peg the Conservatives and Labour in terms of pro- or anti-Brexit. As the BBC pegged these two parties, they were internally divided on Brexit, so the BBC didn't count them toward either pro- or against-Brexit. As Euronews ultimately pegged them, the Conservatives were pro and Labour against. Given Corbyn's' post-election endorsement of a [2nd] referendum, Euronews was perhaps somewhat right in this assessment, at least in the future tense. But the BBC argument that the parties which were indecisive and/or ineffective vis-a-vis of Brexit prior to the election got punished is also strengthened by this post-election change of stance of Labour.
The analysis is slightly different if one uses vote shares instead of seats; the BBC has done the former already (and it's included in your post). The Brexit Party was however more effective in transforming a lesser vote share into more seats than the avowed remain parties. Also of note, geographically, (outside of NI) the Brexit Party came in first everywhere but in Scotland or London.
N.B. here's a compact numerical summary of the election from the Guardian, which doesn't include the Northern Ireland parties in this chart; the latter are 1:1:1 split on MEP seats between APNI, DUP and Sinn Fein; I think of these only DUP is pro-Brexit, the other two NI parties have campaigned against it. The vote share figures are not available for Northern Ireland, because it uses STV.
A video analysis has now been posted by the BBC online, which is a bit more dynamic than what they had before. They also posted a (text) analysis for Northern Ireland. According to that, mirroring results in rest of the UK, the UUP took a beating (lost their seat to the anti-Brexit Alliance) because of the UUP's unclear stance on Brexit.