Are you overclocking the CPU?
While a sudden shut-off or freeze during stress testing MAY point to a problem with the PSU I wouldn't come to this conclusion too fast, especially if you're overclocking.
But if you are NOT overclocking and the system shuts down at stock speeds, yes I would strongly assume that something is off with the PSU.
I am sorry, 450W and 430W does not sound "too convincing" to me and the problem is in addition that the Antec is likely a multi-rail PSU, means it has two 12V rails which are made to draw like 15A each (18A peak) which is about 180W-ish per rail.
NOW - if the PSUs are older it can be that they degraded to a point where they cannot supply the CPU anymore, even NON overclocked. With a weak 430W PSU 10% degradation might be all that's needed....
TDP for a A6000+ is 125W, this does NOT leave a lot of headroom on a rail where 180W would be the maximum and of course not if the PSU is older. Throw in a few drives, fans etc...and you are VERY close to what this PSU can handle.
And by the way, your PSU not only has to supply your graphics card (which is LIKELY on the second rail)....but also your board, on-board devices, pcie slots, sound card etc. and as already mentioned fans, drives and what not. If there is only one rail rated for 15A constant current and 180-ish watts.....uhm....DO THE MATH.... (especially if you happen to share the one rail where the CPU is connected to with the board or whatever else).
This is *interesting* since I just in the recent days read a lot about multi-rail and single-rail PSUs and how *allegedly* multi-rail PSUs are not sufficient anymore for "modern systems".
(This is nonsense, by the way, I am running a Haswell system on a 8 year old Toughpower 750 with FOUR RAILS, best PSU on Earth. I gave a brand-new XFX 750W Core Pro to my wife since I prefer my old Toughpower
BUT: Here is the deal: A weaker PSU like yours has not four rails where ATX board power, PCIE, 12V EPS, power for drives etc. are spread over four rails (which would be fine)...but ONLY TWO rails. And this is where the problem is, IMHO.