Still, it's not like you can go to Indigo Plateau and simply walk all the way back to New Bark, without needing the old man. So, that means it's not just a problem of using the old man's warping option, it's something else...
I dont know how much not having the SS.Aqua II is a problem. Let's see..
When you have the Ghostbuster, you can skip the call and the appearance of the SSA2 and go to Johto via old man or walking back to New Bark, and fix the Ghostbuster (but you'll also skip the Celebi sidequest since you havent called Elm), I dont know if I fixed it for B5 or if I had it fixed only on my version here, but since the SSA2 still doesnt exists, you cant go back to Kanto by sea from Olivine (sailor would say SSA had a problem while traveling to Vermilion), but, you can travel to Kanto using Fly when in the route to the east of New Bark, so you arent stuck. You just cannot travel between Kanto & Johto by sea till you decide to call Elm (and that could be done after doing all the rest of the game, if I'm not mistaken)
What would happen if I still hadnt fixed the Olivine's sailor's script in B5? If you talk to him, you'd go to a broken version of the SS.Aqua, and THERE you'd be stuck.
But if you call him right away, then the SSA2 will show up and not only you are activating the Celebi sidequest but also you are enabling "another" way of going back to Johto, you could still use the old man's method or walk back. The only difference is that when you reach Olivine, the sailor will let you through to go back to Kanto and you wouldnt need to Fly.
As I said, I dont think you really get stuck, it is prefered that you call Elm, though. It'll make traveling easier.