If the question is really, "do I have to lift heavy weights", then the answer is "no", and I, the designated random person on the internet, give you permission not to lift heavy weights.
If the question is "is heavy lifting overrated" or "doesn't heavy lifting suffer from these drawbacks", the answer to both is kind of a factual "no" plus a bit of "maybe you should try a different kind of program".
Let's address it point by point. Your first two points demand the most direct contradiction, but please hold your judgment to the end :)
1. Useful for common tasks
it's not your ability to lift hundreds of pounds for 5 reps...
Improving your maximal strength is often necessary and/or the best method to achieve feats of great submaximal strength. This includes feats of both power and repetition. This is well-established sport science.
2. It's not natural.
At best, this is the naturalistic fallacy.
You did not have weights in 10.000 BC. Our ancestors did not train this way.
We certainly did both. We know that people in ancient civilizations were fans of heavy lifting: are you aware of Scottish stone-lifting traditions, or the Bybon stone a whopping 143kg, supposedly lifted overhead with one hand? The variety of stones like the Húsafell (a ludicrous 186kg), even if not prehistoric, makes it clear that pre-modern people loved getting massively strong (and proving it). We even know that ancient people understood the principle of progressive overload, as communicated in the semi-mythical tale of Milo.
For more, see Crowther's paper, Weightlifting in Antiquity: Achievement and Training.
There is absolutely no reason to expect that earlier civilizations (especially those with metalworking) did not use similar implements that have been lost in the sands of time.
3. Risk of injury.
Anyone who fears injury in the weight room should first become familiar with risks across all sports. Lifting is regularly shown to be one of the least dangerous sports.
Heavy weights lead to higher risk of injury.
I doubt this is true and I doubt the studies one might cite to prove it actually speak to other topics. The greater risk of injury comes from greater intensity, which is present in intense bodyweight training too.
In the end it comes down to personal preference, namely risk/reward. If you want to train light -- whether with weights or bodyweight -- your risk will be lower. If you want more results, you must train with greater intensity, and thus your risk goes up.
4. Requires too much rest.
So what? A lot of people like this.
If you don't, well, it's perfectly effective and sensible to do low-intensity activity every single day alongside heavy lifting. I'm thinking walking, hiking, yoga, ball or field sports, and calisthenics. Or, if you really want, there are plenty of heavy lifting programs that have you working out six days a week, cycling through movements to avoid over-stressing particular muscle groups. If that sounds nice, look into 5/3/1, or "powerbuilding", where the 5x5 approach is combined with bodybuilding to great effect.
Are these slightly sub-optimal for maximal strength? Yes, but no one said you have to train like a superheavyweight powerlifter just because you're deadlifting.
5. Too expensive.
This is extremely up to personal preference.
But if you have the space, a home gym can be gathered inexpensively (in many places) from used gear.
In most places I've lifted, a gym member ship is thirty bucks a month or less. Whether that's overrated depends on whether you see the connection to your long-term health and wellness and how much money you have on hand.
All of this is, I suspect, beside the point. If you want to get strong but hate beginner powerlifting programs, then why not explore effective strength programs that aren't modeled after one particular approach? The three you list are basically the exact same program with different toenail polish.
Maybe 5/3/1's four- or five-day split and variety in rep ranges is different enough for you. Maybe you'd prefer to use strength movements from Olympic lifting (Romanian deadlifts, front squats, overhead presses, etc.). Maybe you're just not suited for these kinds of things at this stage in your strength development and you'll love heavy triples after a year of 20-rep squats or Strongman training. Regardless, I encourage you to look beyond 5x5 training, because that is only one path to heavy lifting.