Inertial frames in Newtonian mechanics and special relativity

Is there a more direct general way to see that (if another frame moves uniformly with respect to an inertial one, then it is also inertial) without computations so that it would work simultaneously both in classical mechanics and in special relativity?

Yes, but it requires a bit of setup and vocabulary. First, for both classical mechanics and relativity set up a spacetime using three dimensions of space and one dimension of time. Note that an inertial frame is then one in which force-free objects form straight lines in spacetime.

Now, if we have two inertial frames then we have the requirement that all straight lines in one frame must be mapped to straight lines in the other. The class of transformations that does this is called affine transforms.

Finally, note that if a frame moves uniformly with respect to another then the time axis of one frame is tilted with respect to the other. This can happen with a shear transform, which maps squares to rhombuses. You can visualize it like taking a deck of cards and sliding them so that the cards remain flat but the stack of cards as a whole is tilted. Shear transforms are affine transforms.

So combining the above if you start with an inertial frame then a frame moving uniformly relative to it will also be inertial because uniform motion is a shear transform which is an affine transform which preserves straight lines which (for free objects) defines an inertial frame.