Are there any Christians who only accept the gospels? Have there ever been historically, or now, Christians who have only accepted either the Gospels, just one of them, or a synthesis/redaction of the four Gospels as being truly inspired or in some other way most basic to being a Christian?

If so, what were their most salient arguments?
Perhaps there are/were others, but the 'Jesus Christians' are the closest sect I'm aware of that matched your description (they are of relatively recent history - beginning in the early 1980s until officially disbanding in 2010: although their website is still being maintained and updated, so perhaps reports of their demise are premature). While not completely denying the inspiration of other scriptures, they placed much lesser value on them than the words of Jesus. 

One of the striking consequences of their distinctive beliefs was the emphasis they place on 'living by faith' by not taking on salaried employment, and believing this is the normative pattern for all true followers of Christ and additionally the positive promotion of celibacy (marriage not forbidden, but classed as an inferior option). 

Overall, I would describe them as a bit like Protestant Franciscan friars except for not recognising any authority beyond the words of Jesus and their groups interpretation of them and being highly sectarian (at least the two Jesus missionaries I encountered in 2005 gave me that impression).