the way the old card system "forced" you to use dps cards was twofold:
1) you had no guarantee to ever see a dps card if you didnt fish for it
2) using other cards would eat up your rr effects

for the old system to work i have two very simple ideas
1) make all 6 cards appear in randomly ordered cycles*
2) make minor arcana let you use a card without using up royal road effects

it doesnt fully solve the problem of rng mitigation being unreliable, but those 2 changes would guarantee you two AOE balance/spear every 180s AND you can freely use the support cards without feeling like youre shooting yourself in the foot.
*redraw kind of becomes a bit problematic since it could let you cycle through cards faster. the way id make redraw work was either just swapping a card with another one in the sequence, or, since youd see al 6 cards every 6 draws anyways, delete it.