In many (possibly most) cases yes, it will give better anonymity, but keep in mind that:
some degree of fingerprinting correlating the 2nd computer to you is possible through the analysis of your keyboard and mouse usage alone (even just mouse movement), and from your interaction with individual or multiple sites, when these interactions use the same Tor circuit
as you hinted at, the other fingerprinting methods will still be employable, they'll just won't be correlatable to what you've done and do through other devices. But all the activity you do through the 2nd computer will be correlatable.
- possibly facilitate hacking
- allow to spoof anything that is being sent to
computer 1, so that for example you might be thinking that you've posted something to a site, when you actually haven't
- allow interception of anything you do through it, including for example the interactions with TLS sites
- through the interception, facilitate your identification (analysis of anything you do with the computer, not just within the browser)
- easily allow to see, even remotely, that it is communicating with
any adversary that can look at the 2nd computer's communications will get to know about the first one
of course, as you mentioned, the global adversary can always correlate
If you don't have physical control of the 2nd computer you don't really know what it's doing: it might well be sending all your traffic to someone, it might have backdoors/out-of band management facilities, and in general it might be very easy for someone else to interact with or spy on, even without any "hacking".
The hacking point means that you should care about the 2nd computer's security too.
The last point is instead actually a very serious threat that this model introduces.
For all your aims except Tor hiding and DNS leaks it would be better, with no drawbacks, to use a computer more under your control.
DNS leaks are easily preventable in other ways (Tor or VPN proxy), so I would ignore them here.
For Tor hiding you should keep in mind the alternative of passing through a VPN (either commercial or set up on a "2nd computer").
This has the drawback that it might be possible to detect Tor usage through traffic analysis (of the traffic between you and the VPN alone).
So ultimately you should balance the threats:
is it more likely and risky that my mere Tor usage gets detected by an adversary other than the global ones or the ones in control of the VPN/2nd computer (who have always other means of detecting it), or that the 2nd computer is accessible to or hackable by an adversary (which allows the interception, in clear, of everything you do through it)?