I now make face frams using pocket screws. The secret to me is making the end cuts precisely at 90 degrees. Tightening the screws them pulls the frame perfectly square. I use a Wixey digital angle finder (new one with AAA batteries) to set the blade at precisely 90 degrees.
If I were doing lap joints as you're using them, I'd also use a dado and shoulder plane. Here's a video on making face frame lap joints on a table saw WITHOUT a dado blade. If you have a glue line blade you'll get a pretty flat surface. The method for setting up the actual cut to get a perfect overlap is simple and elegant. This is probably the way to go if budget is an issue.
https://www.youtube.com/watch?v=Klpn-LutJf8
Theatrical flats are now made using lap joints, and are almost identical to a face frame, but with much wider material, then covered with muslin or a light canvas. The canvas is stretched and shrunk and addes strength and durability so they can be reused for decades. Those joints are made with a router because they are about 4x4 inches in area. The crosscut edges are usually precut with a hand saw to help avoid blowing out the edges of the joint. In a theatrical flat, the joint doesn't have to be as near perfect as it would in a smaller face frame, thus the final fitting with the shoulder plane for a face frame.
Here's a video of making a flat, somewhat similar to what you're doing, but using a router.
https://www.youtube.com/watch?v=Pq5TScibuHQ