Maybe he overspec’d when he upgraded the video card…
What PSU do you have?
I’m always a bit dubious of using old parts with new parts when building a new PC. It’s fine if you do incremental upgrades every year for several years. But using 7+ year old stuff with a new PC (apart from the case and fans) seems penny wise and pound foolish. A PSU might be ok…worth a close look though.
Especially with a PSU, the one part of your system with the potential to wreck every other part of your system, that’s my usual take on it. Old-ass slow memory or an ancient hard drive might not be great but it won’t break anything either.
But I’ll admit that maybe it has a perfectly capable, reputable, PSU in there of sufficient wattage. The GPU had been upgraded in the past so maybe a PSU went with it.
I do share @Jophiel’s philosophy on upgrading the PSU as well, but maybe early next year to help amortize the expense. But only if the old PSU can handle the load in the interim.
Reddit is of the opinion that you can run a 4070 with a 500 watt power supply, though that would feel a little tight for me. Reviewers have tested it with a 550 watt power supply without issue.
However, if I were going to plug a 4070 into a 6-year-old 550 watt power supply, I would want it to be both gold rated and from a reputable brand. I don’t think I would plug it into a 6-year-old 500 W power supply regardless of rating and brand.
Google says for a 1070 you need a 500 watt power supply, so if he overspec’d at all for extra headroom…maybe?
EDIT: In googling around for this information, one of the articles I found mentioned the power connector on the 4070 being the classic 8-pin as opposed to other modern cards using the newer 12-pin.
The RTX 4070 to date is NVIDIA’s most interesting GPU on the market. Its consumption at 200W of TDP can replace all cards of previous generations on any configuration without requiring a change of power supply. Proof of this is that many reviewers with only a 550W power supply tested the GPU without any worries. And as a bonus, it only requires a classic 8-pin connector, far from the confusion caused by the integration of the new 12-pin connector on the 4070 Ti and more.
Almost every time I build a PC for friends or family (work they defer to me) they push back on the PSU. Any old PSU will do…just plug it into the wall. Find a cheaper one. I tell them that while it is not sexy it is the one thing they should not go cheap on. That’s not to say buy the most expensive one but care about it.
When I look at pre-built PCs from Dell and the like you can usually see the PSU is a place they go for deep cost cutting to make the final price more attractive.
I agree with all here…make sure your PSU is up to the job. Also, remember, a PSU should never, ever, be run at is max capacity except for a limited time. Kinda like a car engine or a jet engine…they have a max rated power and you can run there for a few minutes but you need to back off and run at lower power most of the time. So, don’t run at 750 watts on your 750 watt PSU. It can handle a power spike up there but not where it is meant to be. You need some headroom for regular operation.
It’s a total racket, it’s all just the same electricity, right? Jam a couple nails into a Molex cable and plug it into the wall 
I don’t have a 4k monitor but my TV is 4k and I believe I’d be able to do 4k streaming to the Deck through the TV. Is it something I really need? …probably not. I’d rather run things really smoothly at 1080. But I also want to be set for the better part of a decade again.
I’d need to check what PSU I have next time I’m at that machine, but I think it’s a moot point. Sounds like if I try to upgrade my rig I’ll run into a Ship of Theseus situation where I need to change out every single part. I’m better off just upgrading.
Just my $0.02 here but I really think you will see a very nice improvement in visual quality if you bump up to 1440p. I am dubious about 4k. I have a 4k TV and I have seen 4k comparisons on PC but I really think 1440p is the sweet spot for all but the most discerning of viewers and games really made to be at 4k (which are not many, whatever they say on the box).
Remember, going up in graphics resolution really gooses the processing power to get there.
- 1080 = 2,073,600 pixels
- 1440 = 3,686,400 pixels (+44% resolution v. 1080)
- 4k = 8,294,400 pixels (+56% resolution v. 1440)
Getting to 4k requires a LOT more processing power. Power I’d personally prefer that goes into better lighting and shading and textures and whatnot. And 1440 looks good. Noticeably better than 1080 but not noticeably worse than 4k (there are limits to human vision and while you can see this difference the difference is small to most people).
YMMV
How much do you want to spend?
How much are you willing to spend?