Short answer: Nothing.
The thing is, it's not really a "foreign policy blob" thing so much as a shared history of tensions that much of the people of both countries remember, and not fondly.
Plus I think the relationship between Trump and Putin is overstated -sure, Trump likes Putin because Putin is basically what Trump fancies he should be: A strongman whose domestic press does nothing but fawn over him and kiss his ass.
Russia's interests are basically in dominating Europe, because they don't like the idea of being invaded. Understandable, but this is in direct tension with America's interest that no one power dominate Europe, because such a power poses a direct threat (since a Russia combined with Europe would in fact have the resources and manpower to actually pose a direct threat to the ZoCONUS).
Plus the Russians have this mentality of them being the inheritors of the Roman Empire and all the stuff that goes along with that. So...no matter what, even if you fired every other politician and bureaucrat in the U.S. and Russia, it would just lead to them being replaced with other people who don't like the other country.
Good relations between individual leaders won't translate into good relations with countries: Wilhelm I and Nicholas II were *first cousins* and that didn't do shit to stop World War I.
Here's the challenge, have Trump and Putin's personal rapport/admiration actually does something useful for both the American and Russian people and/or the holistic collectivities known as 'the USA' and 'the Russian Federation'.
In OTL, despite mutual admiration and a personal verbal cease--fire or non-aggression pact between the two leaders, the United States, and United States' allies received no relief from Russian cyber attacks. Nor relief from false accusations that the US supported ISIS, and Russia even paid bounties to the Taliban for killing American soldiers in Afghanistan. I'm not sure this last item, a new low, had started yet under the Obama administration. So America's gains from the personal rapport were limited.
Meanwhile, what did Russia gain from Trump's election and administration, in a positive, constructive sense? In the negative sense, it gained an the satisfaction of seeing America being more divided (although President Hillary Clinton would have been plenty divisive, causing many American men to give birth to many a litter of kittens at their shock and horror). But I'm talking about positive, national Russian interests.
Russia didn't get any sanctions lifted, in fact it got a few added, during the Trump administration. Russia didn't get any concessions, or even status quo continuation, on arms control, in fact the US pulled out of the START and INF treaty, in line with neocon anti-arms control orthodoxies.
Neither of them collaborated to leave a legacy of greater stability or peace vis-a-vis each other, by, for example, neutralizing Ukraine (and maybe Georgia) through a modern day version of the Austrian State Treaty.
In the ATL, have these two guys use their strength to achieve something positive & enduring.
None of that is possible. Not because of "neocon orthodoxy" but because personal relationships between leaders are basically meaningless.
Note that Americans as a whole still regard Russia as a geostrategic rival, and have since 1940 or so. Oh, sure, you had a couple of brief periods of detente and whatnot, but the truth is, it was and is a competition. See, Russia has this (not entirely crazy) concern over being invaded, and wants to make sure its borders are secure. The problem with that is, when you ask a Russian just *how* far their influence would need to be for them to feel secure, the answer is "control all of Europe." Unsurprisingly, this upsets the rest of Europe.
And that represents a direct threat to the security of the United States, because a continent consisting of Russian resources and European technology is actually capable of threatening the CONUS. Since Europe has a nasty habit of fighting among itself, it's fallen to the U.S. to not only make sure they don't fuck it up (because doing so would only fuck us over in the long term for a variety of reasons).
The whole "collaborating over peace" and the Austrian State Treaty comment tell me you not only don't understand the current situations in Ukraine and George, as well as what the situation in Austria actually was post-World War II.
Russia *does not want* peace in Ukraine or Georgia, because both of those states are sick of Russia's shit and have been actively resisting Moscow's attempts to pull them back into its orbit since the collapse of the USSR. This is for a very simple reason: Both states want to join NATO, but the North Atlantic Treaty stipulates that any state wishing to join CANNOT have any ongoing border conflicts or disputes, because NATO is a defensive alliance and doesn't want to get dragged into a war (at least not with a country that matters, like, oh, Russia). *That's* the reason for the shenanigans with the frozen conflicts -it renders those states unable to resist Moscow's influence.
The Austrian State Treaty came about because Khrushchev had only recently come to power after Stalin's death and wanted to try and set a new tone for relations with the West. In fact, the West *had* wanted to resolve the occupation of Austria but the Soviets said no, because they knew, given the choice, the Austrians would align with NATO given how the USSR had been systematically fucking Austria over since 1945, and in fact the country only avoided starving to death in 1947 thanks to a MASSIVE influx of U.S. aid, and this was before the Marshall Plan. At the same time, the U.S. wanted to end overt occupation duties (which they were in the process of doing in Germany). So when Nikita proposed to negotiate an end the occupation and let Austria unify without Soviet interference, provided they agreed to stay neutral in any conflict*, the West went "Sure."
And I say "occupation" because that's what we were doing at the time -unlike Ukraine and Georgia, there were U.S., British, and French troops on the ground, which meant Moscow couldn't just fuck around as it pleased. So for Khrushchev, it was an easy win. Note that this wasn't done out of the goodness of his heart -it was as much to remove a thorn from his side as it was to improve relations. In fact it was also around this time that Khrushchev asked if the Soviets could join NATO. Yes, you read that correctly. Why? Because NATO invoking Article 5 requires unanimous consent, and so if the Soviets tried fucking with, say, West Germany, the latter couldn't do a damn thing. The U.S., not being stupid (since the Soviets were doing this with the UN already), politely said no.
*-It should be noted that Soviet war plans with the West actually called for them to violate Austrian territory *anyway*.
Now, as for the New START and the INF treaties you complain about? Well, that's because the Russians had actually been ignoring them for several years anyway. In fact that was one of the problems for Obama when he tried getting the Iran deal ratified -the administration *knew* the Russians had been systematically violating the original START *and* the INF treaties for years, and lied to the Senate about Russian compliance in order to get New START ratified. Naturally, the Senate was pissed. And this ignores the dipshittery with the Russians murdering or at least trying to murder people in Britain and Germany with impunity, etc.
So...yeah. There was never any actual hope of rapprochement. American and Russian interests are just that opposed to make it possible. And as we saw with Wilhelm II and Nicholas II (who *did* try to be as cordial as possible and were first cousins as well), personal relations take a back seat to geopolitics.
Plus, as a side note, I'd add that the whole Trump-Putin "friendship" was/is as overblown as Trump's relationship with Xi. Sure he liked them because they were/are in the sort of position he wanted to be (an authoritarian strongman who could tell everyone what to do), but both of them had no problem flattering him to the point of absurdity because when they did so and then asked Trump for something that would help them and screw over the U.S., Trump would agree because they were "friends". (and then Pompeo or whoever would have to pull Trump aside and explain why that request was not only impossible but actually detrimental to the U.S.).
