No, CR 1 is intended so that the companion does not die in a hit or two. The giant eagle is a bit of a glass cannon that will punch harder early on but will become a bit of a liability in the late game (even with the improvements of Nature's Guardian).
The party might not want to split with the ranger and eagle going off on their own though. Gate's Pass is surrounded by wyverns from the start of the campaign with a nasty red dragon in the area. AC 13 is super easy to hit, and the giant eagle and ranger would come crashing down somewhere on their own, bleeding out (also, Nature's Guardian does not provide an AC bonus). Torrent would point this out (honestly, there is not really a great place in Gate Pass for the giant eagle to have been parked aside from the griffon aviary). This means the heroes need to retrieve the giant eagle as part of getting out of town, or someone might have tried to ride it to fight back during the attack, and the giant eagle was killed off screen during the midnight battle.
The intention was for the PC to have a companion with a bit of survivability that they can bond with over the course of the campaign (and is not useless by the end) while giving concrete combat bonuses that some might feel the original ranger was missing. If you are feeling uncomfortable with that, you might want to talk with the player about generally keeping with the group so that they don't have large portions of story dedicated just to them as well as not being too far away from the party (so they don't get killed).
Another option would be for you to be open with the player about your concerns to see if you come up with a solution between the two of you.
In my campaign, a PC had Takasi, and even early on, the giant eagle had to be cautious about when to engage, and there were some places (like cities) that it just could not go.
I hope those thoughts help. Feel free to ask any other questions or make a counterpoint.