I have both in my system, but I also installed the strainer a long time after the system was up and running, only reason I installed was I added a FPHE for my snow melt system. I have a ball valve immediately before and after so I can clean it and only lose a minimum of water. Some recommend a bypass around it so you can leave the system run while cleaning it, not sure if this is really required if you check it several times after the first start up in the shoulder season. I’ve yet to find anything in the strainer, but again my system ran a long time before I added it.
I’d highly recommend adding a fill valve in your basement, nothing like having to drag a garden hose in the dead of winter to top off your OWB.
While you have the trench open for your underground pipe go ahead and lay another pipe for water by your OWB, I did this last year and should have done it 15 years ago on the first install.
When I redo all my plumbing in the basement I plan on adding a solenoid activated fill valve, I already have the extra wire ran to the OWB and will install a pushbutton there to activate the solenoid to avoid overfilling and losing treated water, which can happen when you have a manual fill valve in the basement.