I remember you said you weren't American, so perhaps you will find this enlightening:
in 2002 the Democrats hyperpoliticized Paul Wellstone's memorial after he (an arch-liberal niggerfaggot who started out as a community agitator for the far left in the '70s and was running for another term in the Senate at the time) died in a plane crash shortly before that year's Election Day. They booed the Republicans who attended, aggressively tried to solicit votes for him, promoted his last policies in life, basically went the whole nine yards to turn his funeral into this ghoulish parody of a political rally.
Voters were so disgusted by the Democrats' behavior that the Republican candidate who originally faced Wellstone, Norm Coleman, flipped a -8% deficit in the polls into a +2% victory over Walter Mondale (former Democratic presidential candidate & himself a longtime Senator from Minnesota too) after the latter had replaced Wellstone on the ballot. So yes, as a matter of fact it is entirely possible to overdo wielding a martyr's corpse as a cudgel in American politics & end up alienating the very same voters you were trying to rally, the Republicans already took a risk as-is with how they handled Kirk's memorial and I would definitely argue (based on, for example, the results of recent primaries where the candidates supported by the administration present at said memorial won) they managed it better than the Democrats did Wellstone's.