Carding people should take you 0 extra seconds because you can
Spell1 (Play) > Spell2 (Draw) > Spell3 (Play) > Spell4 (Sleeve Draw) > Spell5 (Play) > Spell6 (Play) > Spell7 (Play)
in the same time it takes to
Spell > Spell > Spell > Spell > Spell > Spell
AST's cast times are shorter than the GCD for specifically this purpose. You can lose about 0.25s if you try to card after a BeneficII, Helios, or Aspected Helios, but in most cases if the accumulation of 0.25s concerns you you can either Lightspeed or just delay your card for one GCD so that you're playing it after whatever shorter spell you cast next.
All those Spires are the cards telling you you should have brought a ranger or caster. Regardless of whether the party was premade or matched, that'd still be true.