IMHO, 'thinking like a drummer' is very important.
Take Latin Music...pianists, horn players, bassists, & even guitarists are 'thinking like a drummer/percussionist'.
Interesting comments & approaches here...
Gary Willis sez the hi-hat is important 'cause it's the most consistent component of a drummer's groove(on-stage, that's where bassists should be; next to the hi-hat).
There are a couple guys I play with where it's the ride cymbal....thank God it's doing quarter notes(1-2-3-4) 'cause everything else is 'out'.
While I think it's important to be able to play the kick drum's pattern on the bass...it's also very important to be free enough as a musician & move/shift/displace the bass OFF the kick. That is, 'fill in the cracks'. You can check out some minimalistic Latin tumbaos &/or Reggae bass lines as examples.
And-
Trust me, some drummers will move the kick into the backbeat...then whatdayado?