The only hard DPS "check" fight anyone might be hitting is E2S and that is more so of can you dodge heads during Quetas phase than just being good at DPSing. Leviathan? Its an entire party wide mechanic check, yes the DPS requirement is tight if you are near the minimum ilevel but most of the mechanics are full team wipes rather than single target screw ups.
Titan... Titan is a marathon fight, its clearly made to be a 'training mode' for Ultimate and that is something we honestly needed. Omega wasn't that hard besides just "Hello World" but once you got it down the fight was really straightforward.
If you are somewhat decent at keeping uptime and understanding your job then you shouldn't have any issues with DPS checks unless its week 1-3 and you are on the 3rd or 4th fight already.
If you are going to blame your healers for not doing enough dps... well that dpeends if they aren't doing a bare minimum of casting DPS spells when no one is taking damage, you don't need to be a 'pro' at managing your heals and DPS... you just need to be competant.
Tanks....
as a Tank main there is almost no reason why a tank shouldn't be dealing a good amount of damage. Besides GNB none of the tanks rotations are difficult or very intensive so you should be doing adequate damage during the fight. All you need to know is "what do I need to do for X mechanic" and "tankbuster/cooldown management". There is very little thought that you need to do about that once you experiment with cooldowns spaced out. No excuses for tanks not DPSing anymore, tank stance no longer hits your personal DPS and you shouldn't need to even worry about hate. This isn't SB anymore where tanks can hide behind "hate management" or "tank stance". In SB bad tanks sat in tank stance the whole fight, you are a blue DPS, focus on what you need to do, then once you know those requirements.... think like a DPS.