Not necessarily. Bats for instance evolved flight separately from birds, and can still climb.
Evolution will try everything. If there's sturdy carnivorous rats around, the rest of the rats will try their niche. Some will try faster, some will try playing dead, others will dig holes to hide in, or climb trees. Some might grow thicker skin, or looser skin, or venom, or just being poisonous to eat.
These empty niches can be filled pretty quickly. After removing top predators like wolves for instance, the average size of coyotes has been steadily increasing over the last 50 years.
A side note, plants figured (in the evolutionary sense) that birds were great seed spreaders compared to the non-birds. Some evolved impressively strong spices (Capsicum) that birds are impervious to (up to several % by weight) that eliminated non-bird consumption of their berries. So to protect bird feed from squirrels at Capsicum.
Any bird like feature will help with a particular niche. Feathers, beaks, dexterous claws, bird songs, gliding, and of course powered flight. So it's not like you need a huge jump from land based mammal to full flight before you have any evolutionary advantage for a particular niche.