Of course, if someone wants a specific servant, story-locked or not, that’s the primary consideration.
But for F2P, the odds of getting a specific general pool servant as a random pick from a banner is so small it should basically be not a consideration.
For example, the odds of getting a specific SR when there’s no SR on rate-up is probably around 0.06% (I’m using the story summons odds since I don’t have an example atm for such a banner–it would probably be a little greater as the handful of story-locked servants wouldn’t be there to dilute the pool as much). This is equates to 1 in 1667, so maybe as high as 1 in 1500.
But if there are any SRs in the banner, these odds tank. In the current Da Vinci banner, there are 2 SRs on rate-up, and the odds of a specific non-rate-up SR is just 0.017%. Therefore the odds are just 1 in 5882.
Of course, if the one wanted is on rate-up (e.g., Emiya is now at 1.2% or 1 in 83), and if one was expecting to roll heavily on such a banner, then that’s a bit different. But still, as people have rolled 1000 for an SSR and not gotten them, nothing’s ever certain so I would still say that hoping for a lucky SR drop is not something I’d ever make a consideration between two servants I was considering, at least with the rate that I roll–which I would have to guess is in the 600-800/yr range