Cotr has a few things going for it:
1. It is twice as space efficient as other $2 villages. If you are aiming to draw deck, you need only draw one Cotr a turn to get the effect of two squires. This makes a lot of +3 card engines more viable with less support (trashing, sifting, etc.), it is still inferior to vanilla Village for a lot a space efficiency concerns so I have seen a lot of people overbuy it.
2. It is exceedingly reliable. It is totally worth it to build your deck off some other village, but add a CotR just to reduce hands where you whiff with an all terminal draw. It is particularly nice on high draw/low trashing & sifting boards for this.
3. It can enable turns full of Terminals with no/very limited draw. E.g. I did a Bridge, Haven, Doc, Iw, Cotr deck where I used Havens to get 6 Bridges & 2 Iw in hand. These boards are rare, but fun.
4. It interacts very well with everything that likes Treasures. Mint is insanely nice - trash coppers, Mint village; Smithy/Mint/Cotr is pretty reliable and very nice with almost any +buy. Mine, Forager, Taxman, Stables, etc. Bm and Storyteller are exceptionally good as you can buy fewer CotR being able to fully cycle the cards through the Tavern Mat with just one pass.
The downsides:
1. It cannot provide +actions when gained. This makes it much harder to ramp up turns with cards like Iw or Workshop if you villages are delayed a turn. One of the most powerful things for an engine is the ability to gain & play cards the same turn. Cotr takes an extra turn to get to the equivalent point of another village.
2. It reacts poorly with cards that care about actions. Most notably Kc, Tr, Prssn, and Rc. Others are trivial (like Transmute), but you can have problems with things like Vineyard, Library, Graverobber, Peddler, and Conspirator.
3. It is makes you engine highly vulnerable to Thief and like any $2/$5 village to Swindler.
4. It is not as space efficient as regular village. Yes having one or maybe two is great, but if you are drawing your entire deck it really is inferior to most any $3 or higher village.
A few specific interactions:
1. It plays very nicely with Shanty town - you use the actions to clear out terminals (particularly non-drawing ones) to play Shanties as activated cities. This is particularly nice if you want just one or two non-terminals in your Shanty deck (e.g. a Militia and a Masq).
2. Golem will skip Cotr, but it makes it much easier to manage actions when you can leave spare actions on the Mat when your Golem chain leaves you with unneeded actions.
3. You can spike an end game province very easily with Cotr and Diadem. Purely a tactical play where you already went Tournament/Cotr and just happen to line up Prov/Tour and two Cotr on the mat (Tour draw & coin must not be blocked, return the Cotr after resolving the Tour).