Higher intensity, which doesn't have to come from heavier weight per say, but increased power, work/time meaning total work, higher weight up or time down.

HIIT/interval/crossfit style are prime examples of in my opinion the most efficient way to achieve "exhausting" workouts (lowest time commitment, and with group gym style, lowest mental effort to formulate workout plans; it's all left to the trainer).

I’d also suggest some interval training. If you have access to stair steppers, I highly recommend them. Easy to get started, and not too tough on your joints. 2 minutes at a pace that’s tough to sustain, and then 2 minutes off. I have a max HR of ~200 and I tend to level off at 180-188 during the interval section. Once you get accustomed to it you can play around with the timing ofc.

Watch out for that spikes your adrenaline like sprinting or heavy lifting in the evening though. That said I always get amazing sleep after heavy deadlifts as done properly they fatigue your CNS much more than the individual muscles so its like your brain forces a shutdown to recover. Squats and bench just wake me up though.

I would think that more strenuous cardio would be more effective than lifting heavy.

I've found certain types of martial arts to be one of the most exhausting activities I have ever done. Boxing, Muay Thai, BJJ, Wrestling, no amount of lifting has gotten me as exhausted as those have. Slept like a baby.