Is daisy-chained wiring at my outlets actually a problem?

Pass-through wiring on the screws is completly normal and code-legal, as long as there's only one wire per screw, which sounds like what you've got. It's not the hack job some forums make it out to be. The reason people prefer pigtails is that if a receptacle ever gets pulled out of the circuit (say you swap it later and don't reconnect the feeds), everything downstream stays live. With pass-through, taking that outlet out of the loop kills the whole run past it. So to answer your question: yes, a loose connection there would drop power to everything after it, and no, it's not automatically a fire trap. A loose screw terminal ussually just makes itself known as flickering or a dead circuit before it becomes dangerous. The failure mode people worry about is a bad connection arcing for years inside the box, and that's more of a backstabbed-wire problem than a screw-terminal one. Those old outlets with worn springs that won't grip plugs are worth replacing on that alone, so you did the right part already. Since you've got boxes open, turn the breaker off and test dead before touching anything, which you probly know. On materials: late 60s wiring is almost certainly copper, either 14 gauge on a 15 amp circuit or 12 gauge on 20. Strip a bit back and hold it against a wire gauge card, or take a sample to the hardware store and compare. Buy both red and yellow wire nuts, they're a couple bucks, and you'll have the right size whichever gauge it turns out to be. You don't even need a spool; most stores sell short lengths or you can cut pigtails from scrap. Honestly though, if the connections are tight and the terminals aren't scorched, I'd just do pigtails at the boxes you already have open and not chase every box in the house. The ungrounded outlets are the thing I'd actually spend energy on. A 3-prong receptacle without a ground should either be a GFCI marked 'no equipment ground' or stay 2-prong, beacuse that's what protects you, not the pigtail job.

One thing I'd check before buying anything: late 60s is exactly the window when some houses got aluminum branch wiring, so look at the conductor where it enters the box. Copper is that dull orange-brown color, aluminum looks silvery all the way through, and sometimes the jacket is stamped AL. I found this out the hard way in my own place when I bought a pile of connectors and then couldn't use any of them. Aluminum needs special rated connectors and antioxidant paste, and regular wire nuts on copper-to-aluminum joints are a genuien hazard, so identifying it first saves you a return trip. If it turns out silvery, stop and read up before twisting anything together.

The thing that made this whole job painless for me was a plug-in outlet tester, the little three-light kind. Costs almost nothing and it told me in ten seconds which outlets were acutally grounded and which were just wired wrong, no crawling around with a meter. Turned out half my 'ungrounded' spots were just an open ground at one box upstream, and the tester's lights narrowed it down fast. Also worth checking the screw tightness while you're in there, old devices can be loose without looking it. Give each terminal a firm snug with a screwdriver, hand tight plus a bit more, and you'll sleep better.

Pigtailed the whole first floor of my old house a couple summers back, nine boxes total, and once you get a rhythm it's maybe fifteen minutes per box. The trick that saved my sanity was doing one box at a time and snapping a phone pic of the screw layout before pulling anything loose. If you get distracted halfway through or come back to it three days later, you're not guessing which wire went where.

Also have a look at the back of those old devices while you're in there. Some from that era have the push-in holes as well as the screws, and it's amazing how often one conductor is sitting in a hole instead of under a screw. Grab each wire with pliers and give a firm tug, if it slides out it was stabbed, not terminated.

Update: got through four boxes over the weekend. First thing I did was check for aluminum like one of you suggested, since this house is right in that era, but it's all copper so no drama there. Pigtailing went quicker than I expected once I got going. Took a photo of each box before touching anything and turned the breaker off for every one. Outlets all hold plugs tight now.