tere are good explanations for most of them anyway.
There are explanations now that we’ve had ages to ponder them. Conversely, the novelization for TLJ isn’t even out yet.
The TFA novel straight up explains how the Starkiller beam could destroy a planet on the other side of the galaxy from where the base itself is located. It just depends on it you want to accept that explanation.
That is to say, you like the movie or you don’t, and that’s your choice. But holding ST and OT up to different yard sticks isn’t exactly fair.
We shouldn’t need novelizations to justify the events of a movie.
But as for different yardsticks, the Death Star was a moon sized space station with essentially a big version of a blaster that could blow up rocky planets. It required the resources of a galaxy-spanning empire to build.
Starkiller Base is a piece of construction many times larger than the Death Star, with a primary weapon requiring seemingly universe-breaking technology that has never been previously hinted at or explained, built by an organization that by all indications is a fraction the size of the Empire.
These are not two yardsticks.
Iteration is your answer.
The German Empire was defeated in WWI, the Nazi’s “rose from the ashes” and 20 years later the Third Reich invaded Poland and WWII began.
The engineering iteration upon the previous weapons, saw the war machine now employ cannon that could span the English channel, unmanned V2 bombs, U boats, and any number of other more advanced hardware (including ultimately nuclear weapons).
Pretty obvious that the First Order is based upon the hardware of the Empire (TIE Fighters, Star Destroyers, Stormtrooper armour, Starkiller Base, etc), and so iteration is your answer to why they are more advanced.
But that still doesn’t answer the question of why they were able to build a far more ambitious project with far less resources. If we saw that they used a robotic workforce and had a lot of automation for their fleet it would make sense, but we get no indication that it’s different from the Empire in this regard. Hux even says that it’s a machine ‘that you have built’. Yet another missed opportunity if you ask me.
Well, how did the Empire do it? They built two huge battlestations in secret. The First order built a superweapon on a small planet. Not quite the same level of construction required. the weapon is more powerful, but the setup is not larger. And we see this one fleet. How many fleets does Snoke have? How many fleets did the Empire have? We see and increase in the technology, but we don’t really see a larger force. The Empire had to dominate the galaxy and keep the rebels at bay. The First Order destroyed the Republic capital and fleet (one fleet in one system, but evidently all the Republic thought it needed) so they don’t need the huge resources the Empire had to have the fleet and ships we see. When you move beyond the movies, the Empire was massive and there is no indication the First Order is anywhere near that big.