Hi
The leads that come with the ut139c should be perfectly ok.
I used the Tenma rebranded version at work and the leads are perfectly useable.
I personally have 3 Uni-trend dmms:
Ut70b (Tenma rebranded version)
Ut61e
Ut210e
The leads for all of them are actually ok. Slightly better than cheap ones you get from Ebay/Amazon.
Yes, if you want you can replace them with better quality leads. Try the Probemaster ones people here on the forum recommend.
As I said, I took a chance and tried a probe kit from Amazon, like the one you suggested, but unless someone has tried it, it is a gamble.
The one I got seem to be pretty decent. More flexible than the UniT leads, probably the same as the original Fluke leads I have. They seem to be gold plated but like all cheap leads the gold plating is not that great. Not sure I would trust them to handle 10Amps.
I bought the kit because I needed new leads to go with the Agilent u1252a I got and wanted leads with alk the accessories. I am pleased with the kit I got, the leads are perfectly useable.
Edit: just looked at the kit you asked about again. I am not a fan of the probes that you plug onto 4mm banana plugs. Agilent/Keysight use that kind of probes and I personally do not like them.
I would definitley avoid the ones where you screw the tips into a body. They will wear and become loose. It is ok to have accessories that clip over the tips like insulated large crocodile clips but avoid probes where the probe tips screw in.
I usually look for probes that are moulded on the end of the leads. I have lots of the cheap ones from Amazon and they are useable.
If I remember and I can find the insulation meter at work, I will take a pair of the cheap leads into work and test the insulation. We have a insulation break down tester as well (goes up to 6KV), could try the cheap leads on that if I can find someone to supervise.