Can I solder a RG316 coax directly to a PCB instead of using the intended SMA connectors?

Soldering directly to the pads of the SMA connector instead of using SMA connectors should work fine electrically.

The obvious problem is that the cable is no longer removable. If you're fine with that, then go ahead.

It would be good to have some mechanical strain relief if this has a chance of getting flexed. At only 2 inches wavelength, you do need to be careful about impedance changes over only a few mm. I'd stay away from adding any material around the solder joint or the wires outside the shield. However, you can do most anything you want back down the cable a bit from the solder joint and where the shield is still intact. Perhaps you can hold the cable down with cable ties or something further from the board.

Excellent advice in comments and answer thus far. Especially, yes I agree, nothing over the soldered joint itself. BTW, such a beautifully soldered joint as shown, IMHO, shouldn't add any more loss than the physical dimensions of the proper connector would otherwise do.

I would only add to please obey any advice you may come across from the providers of RG-316 cable. It is in my mind that now that you no longer have a large-ish connector on the end of the cable, there may be a temptation to thread this small diameter cable when re-installing, through nearby holes or bends which may kink it or put a sharp bend in it while also providing some strain relief.

Please, definitely no kinks or sharp bends in coaxial cables carrying RF. Do any strain relief you wish to, but nothing tighter than a gentle curve in the coax or you will lose RF in the cable.

Personal opinion: if I was doing something similar to any equipment of mine, if it came to a choice of kinking a cable or not having strain relief, I would definitely choose no strain relief. I mean, how rough do you plan on being with your own equipment?

Bear in mind that a cable tie is a good idea for strain relief, but if you pull it tight so it squashes the cable even a little, you will have altered the characteristic impedance right at that place in the cable, and you will lose RF due to setting up circulating currents by creating that very discontinuity.