Warmtone papers are warm traditionally because they use a finer grain emulsion and typically use more chloride halides vs bromide. However, I believe this is mostly a moot point on modern emulsions. The warmest paper I know of still produced is Fomatone Classic Warmtone FB. It's really something special and can still produce brown and even reddish tones in specialized developers, and gives warm olive tones in normal developers... However, everything else is just meh in my experience. The warmest warmtone you typically can get is olive tones, no browns and definitely nothing close to red.
I did a ton of research and formulation tests toward this problem, even making my own emulsion and messing with salt prints. My conclusion is that getting warmtones into browns and reds requires a very specialized emulsion (even very fine grain 100% chloride is not capable without extra effort) to shave the grains down even finer than is possible by home means, while also adding dyes etc to prevent hour+ exposure times in contact printing. So given this and the state of the industry today, all hope is lost right? Well, maybe, maybe not.
Theres a few processes that will produce browns and reds. The most accessible of them is lith printing. The exact chemistry of lith printing even today is not 100% known, but the likely overview of how it works is that it works by the following:
* First there is the "induction period". This will produce soft and very warmtone highlights. It does this by several mechanisms. Potential mechanisms include: polyethylene glycol coating silver grains, rendering them permanently undevelopable or maybe just permanently impartially developed, hydroquinone + bromide effectively "bleaching" the latent image, hydroquinone simply being incapable of penetrating silver grains so only the thin surfaces of the grains get developed.. etc
* Second, the infectious development period. Basically in places where a lot of grains survive and are developable, eventually the hydroquinone takes off in a series of reactions creating radicals which are effectively very short lived fogging developers. These radicals are so short lived that they only really get a chance to accumulate and do any real development where there are a lot of grains to develop (ie, the shadows)
Note that lith printing can also work after bleaching, typically producing much warmer tones than doing it "first pass", and especially when using a copper sulfate + chloride based bleach
Most old warmtone developers defy the common "guidance" to develop to completion. You can confirm this yourself by using a film developer and pulling after just 1-2m, you'll have brown tones but practically no black depth. These developers and this anti-guidance do not work on modern papers though. You'll get very poor contrast and lack of black depth. I'm unsure what changed about modern papers. Many people point to cadmium salts, but many highly experienced chemists say this was not the case. I personally believe it's addenda that was added to make papers more "fool proof", including anti dichroic fogging agents, anti-fogging agents, incorporated developers, and emulsion hardeners. Vintage papers if you've ever worked with them are a lot more fragile than what we work with now.... but honestly it seems like no one really knows what changed. On the otherhand, trying to produce my own emulsions I'm not able to get them to work either, likely due to the emulsion being too primitive.
A more traditional process I've discovered though is using a specialized developer which will basically eat at the silver grains and will keep them in solution (NOT redepositing them, causing dichroic fog). I'm unsure of the exact mechanism, but thiocyanate and thiosulfate will not produce this effect, nor will massive amounts of sulfite. I believe this is because thiocyanate and thiosulfate will "break" the grain allowing them to be more deeply developed into silver metal before the silver halide can be carried away. In my tests, even large amounts inducing dichroic fog will not give warm tones, and rather will just cause black tones to be deeper and also etch away at highlight details. What does work to produce this effect however is ammonia, or in the more easily accessible ammonium salts such as ammonium chloride and ammonium bromide. I believe this works by rounding off developed silver metal, maybe forming silver diamine, while also dissolving silver halides without actually "breaking" the grain allowing it to be more easily developed... However, I've noted that "seasoning" with such a developer is essential, some silver in solution is essential, so maybe it's dissolving and redepositing finer grains of silver somehow?. Ammonium salts decay in alkali to produce ammonia in solution, which easily evaporates. In other words, the effect will be somewhat short lived (tray life of hours, shelf life of days) and the developer will smell of ammonia... Meanwhile, you're also fighting against dichroic fog that is easily produced and trying to ensure that enough silver will be present and developed to produce a good black tone.... Despite this, it's definitely possible to formulate such a developer.
My custom developer GVPX2 will produce such results and is specifically tuned to be as shelf stable as possible. It lasts for a few weeks in a bottle, but will only last a few 1 hour long sessions in a tray before black tones decay too much and the brown warmtone effect vanishes, giving boring olive tones instead. I've not tried to improve the formula much, but I'm sure it could likely be much better. Either way, here it is:
700ml water
"pinch" of sulfite
1.5g metol
30g sodium sulfite (kept low to prevent too rapid of metol regeneration and solvent/replating effect)
5g hydroquinone
50ml triethanolamine 99% (TEA, sourced from photographer's formulary, unsure if critical here)
15ml glycerol (used to reduce ammonia smell and keep ammonia in solution longer)
0.15g benzotriazole (can be increased to slow the developer without otherwise affecting results)
0.5g potassium bromide
1g ammonium thiocyanate
12g sodium metaborate (carbonate will react with ammonia)
8g ammonium chloride
Top to 1L with water
Final pH ~9.75
Exposure should be slightly increased and the print pulled before complete development, typically 2-3m
Example:
https://i.imgur.com/I5Z0mFN.jpg on Ilford MGV. Biggest problem is that shadows have fairly drastic reduced contrast (indicating too much "highlight compensating effect" in terms of negative developer), but in person black depth is pretty good and this image definitely matches the daylight appearance