You're missing the forest for the trees, and they hint as much as well. No singular piece of evidence can be used as proof of an ancient civilization because it's likely too difficult to distinguish from natural lines of evidence of an identical nature; however, as I said earlier, and as they say in the paper, combining several lines of evidence can distinguish from natural events such as carbon isotope excursions with a large long-lived radiogenic isotope excursion (^(244)Pu and ^(247)Cm), and perhaps plastics as well.

I might add that the paper doesn't really dive into quantity too much with stable isotopes, though I believe it could benefit from that analysis. What I mean by this is that while they discuss abrupt events (hyperthermals), they don't really go into abrupt abrupt events. For example, while they discuss ^(13)C excursions such as the during the PETM they don't note the difference in carbon flux between that event and modern day climate change. For example, the PETM released roughly 3000 Petagrams of carbon over a period of ~6000 years. It is hypothesized that under a worst-case-scenario our civilization could release 5000 Petagrams of carbon over a period of 500 years. There is nothing in the geological record that even comes close to those values on global scales. On a yearly basis the PETM rates range from about 0.3 to 1.5 Pg C/year. For comparison, the current rate of carbon release to the atmosphere is nearly 10 Pg C/year. And of course the more lines of evidence one uses the more clear the picture becomes.

As for Mars, I'm not quite sure what dating techniques become irrelevant. Absolute dating methods would work just as well on Mars as they do on Earth, and Martian meteorites as well as lunar samples have been dated using radiometric techniques.

>...two scientists (one is a director of NASA’s climate studies and another is an astronomer and physicist, IIRC) argued that there could have existed an ancient industrial civilization on Earth long enough ago that we wouldn’t be able to gather sufficient evidence of its existence due to limitations of dating techniques.

That's not what they argued. The take away was that any ancient civilization would be detectable through large, abrupt, isotope excursions (anomalies). For example our civilization will eventually be compacted into a thin sedimentary layer; however, stable isotope anomalies of carbon, oxygen, hydrogen and nitrogen will be detectable - discerning them from natural excursions may prove difficult though other biomarkers would help shed light on the full story. Plastics, fossils, and radioactive isotopes in conjunction with the former (among a few others) would surely be a clear indicator of an ancient civilization.

You can read the full study here: The Silurian hypothesis: would it be possible to detect an industrial civilization in the geological record?

I'm not sure what you heard exactly but the reality is that unfortunately it's simply not possible.

Setting aside the difficulties of transporting the ludicrous amounts of nuclear warheads required to heat that much rock, how would you deliver the nuclear warheads to the required depth even if you could get them to Mars?

The deepest hole on Earth (Kola Superdeep Borehole) took about 21 years(?) of active drilling to reach a depth of ~12 km. At that point drilling was too difficult to continue as temperatures were hotter than expected and the rock began to behave like a warm plastic so the hole would collapse on itself (on Earth there's a transition zone where rocks go from behaving in a brittle fashion to behaving in a ductile fashion known as the Brittle-Ductile Transition Zone, this transition will exist on every rocky planet at slightly different depths). To reach the core of Mars, you'd need to drill down ~1,560 km. That means that on Earth, the Kola Superdeep Borehole managed to reach an equivalent of 0.7% of the way to the Martian core. Just over half of 1%.

Hopefully that helps paint a partial picture as to why the idea of restarting the magnetosphere on Mars has no basis in reality.

That's not how self sustaining geodynamos work

It's more to do with size, as smaller bodies will loose their heat to space more rapidly than larger bodies. However, even this assumption has led to some surprises. The moon, for example, was once thought to be solid - far too small to retain any heat today - but recently was found to have a fluid outer core and mantle with partial melt. Ultimately, as magma rises due to its buoyancy relative to the surrounding mantle rock, the moon may actually become volcanically active in the future if the partial melt continues to rise upwards through its mantle before it solidifies.

Research Paper: Tectonics of Cerberus Fossae unveiled by marsquakes