One thing to keep in mind is that regardless of “interference”, doing more work requires more recovery, no matter what you are doing. This is especially the case when the work you do directly causes muscle damage - strength training is a different beast than aerobic exercise and combining them means more recovery. Let’s say you don’t do any endurance exercise, only lift, you certainly cannot do legs at the gym every day. Twice is ideal, some people do three days, but most people end up injured doing legs three days a week. The point I’m making is that I don’t bother with interference I already know that strength training recovery and adaptation takes a lot of time even if you aren’t doing other exercise. Adding endurance exercise means more recovery time because it’s more work you just have to accept that and plan your training accordingly. In your case - running - you are dealing with eccentric contractions and impact at foot strike that causes microscopic damage, so even more recovery than you would need if you were lifting and pedaling.
Regarding weights, I would recommend doing both Bulgarian split squats and single leg leg press. Both are unilateral, both work quads and glutes, but the BSS develops balance and knee stability while the leg press, due to the fixed path, allows increased weight and thus greater force production. Doing both is very complementary. As single leg exercises I would not go below 8 reps (higher weight can strain the knee and, during the eccentric phase, can risk a hamstring or glute strain) and use this regimen: most weight you can use for 8 reps (leaving 1-2 reps in reserve) with a slow and purposefully controlled eccentric phase, increase the reps over time (weeks/months) until you get to 15, then drop back to 8 and increase the weight a bit, repeat. Distance and hill running benefit more from increasing the fatigue resistance of fast twitch 2A fibers than learning how to recruit greater numbers of 2x fibers and this will accomplish that. I’m not saying only do these, there are plenty of bilateral exercises with greater weight that you can/should do, just offering these as good ones for running.
If you decide to experiment with endurance exercise after lifting (same day, not necessarily immediately after), you might find cycling or stair climber to be easier to tolerate and recover from than running since there isn’t additional stress from eccentric contraction and impact, and you still get cardio time in. Or you might find that what works best is running after lifting one day, stair climber or easy pedaling after lifting on the other. Gotta find what works best for you. Regardless of the form of aerobic exercise after lifting, you have to have your dedicated longer running days, and day to day and week to week your legs might feel terrible, but this isn’t about being ready in a day or in a week, it’s long term, both 35 weeks out and months/years thereafter. Every week should have one or two recovery days, and every so often you should have a recovery week.
FWIW, IMO the treadmill should only be used if you cannot run outside. In that case it is clearly better than not running at all. The issue I have with it is that the motorized, posteriorly moving tread reduces the amount of force your leg muscles have to produce to maintain a given pace. You’ve probably seen people who set the speed at 10 mph (6 min mile) and their legs are flying for the next 20 or 30 minutes, but they can’t last more than a minute or two trying to run that pace on the road as their muscles have to do a lot more work than they are used to doing.

