I put box royals in a k320 a while back, didn't like the cherry blues that it came with
there's no way to change the actuation point on a switch without modifying the switches (ie changing out the sliders) unless it's a hall effect switch, which speed silvers aren't.
you can solder in any switch you like, so long as it's cherry mount. desoldering the existing switches is a bit of a pain if you don't have the right equipment and you may damage the circuit board.
gateron phantom browns are compatible with the k320
you can just install MX browns but I do recommend lubing them. the phantom browns are pre-lubed and don't need any help, so I recommend going with those if browns are what you're after.
edit:
Welcome to geekhack
Mechanical switches are digital - either they are pressed or they are not so you can't adjust the height in software. Topre keyboards are not mechanical so you can adjust the height of the keypress on them.
Nearly every switch you see for sale new is "MX compatible" because Cherry had the patent for MX switches and only once the patent expired did everyone start to make copies, if it has a + shaped stem it's going to be one of them.
Gateron are certainly MX compatible so they will work in your board.
Thousands of keyboards are sold with unlubed switches and their owners did not die, so you could certainly just install them. Some people claim to feel a huge difference after lubing, others would say it's confirmation bias that for all the effort they put in they expect it to be better, so it is - the reality is usually somewhere in the middle. Have a read around whatever switch you do chose as some are more scratchy than others and this is what lube reduces.
Another thought - you say your board feels basically great and it uses MX silver which are 45g linear switches, but you're looking at Gateron brown switches are tactile and heavier at 55g. I'm not familiar with the phantoms but they call it brown so chances are it's only a small tactile bump, but it's a bump and will feel different. If you like the smooth feel of the silvers you might want the ghost yellow (which is the same stiffness as the brown) or red which is the same as the silvers.
re: your last paragraph
I suspect the big reason OP doesn't like the speed silvers is the short 1.2mm actuation of the speed silvers. the short actuation threw me off when I got some gateron optical yellows a while back, but I got used to it pretty quickly. any normal 2mm actuation switch should do the job here.
the gateron phantom browns feel similar to cherry browns, but the tactile bump is somewhat more pronounced and they are much smoother.