Is cutting a DIY spline channel around the openings of our wood screen porch a good plan?
This is a good idea, except 1/8" x 1/8" is kinda small. 5/32" would let you use common .160 spline. Check spline availability in your area prior to choosing groove size.

Yes, "trying to use the circ saw on a vertical surface is a bit dodgy". You should remove the panels and lay them flat on sawhorses if possible.

You should consider using a router, which would be easier to control (i.e. perhaps safer). Another benefit of the router is that it "does corners". With a circular saw (or even a table saw) you will need to saw beyond your spline corner to achieve the correct depth at the corner, which will look crappy if the cut extends beyond the wood molding.