Any heat source exposed to space has the basic problem of its volatiles running away and not coming back. On geological time frames, this effect is probably nearly instant, so it isn't very helpful that maybe a new volcano could pop up somewhere and melt lots of stuff or something, because all that does is put a hole in the permanently frozen layer. Any heat source not exposed to space might harbor life, but that's not particularly true of these planets; we already have things like Europa locally with the same possibility.

The issues with valuable resources running away and not coming back is a serious one even for Earth, which I think might surprise some people. The geological carbon cycle [1] in particular is one that may surprise people who have not encountered it before, or not thought about it in terms of extraterrestrial life. On geological time scales it's surprisingly easy for entire important elements to go find themselves a low-energy configuration they like and disappear from any putative biosphere. The issues with a tidally-locked planet providing such solutions to "all gasses liquids" is merely an extreme example of the case, and really makes one wonder how it would be possible for such a planet to stay "stirred" enough for life to have access to what it needs to develop.

[1]: https://en.wikipedia.org/wiki/Carbon_cycle#Geological_carbon...

The link you provided suggests that one of the main ways for carbon to enter the atmosphere or ocean is through volcanic eruptions and geological hot spots. Isn't increased geological activity one of the results of orbital resonance?

If the 'central' planet has 6 close neighbours of equivalent size in resonant orbits would this not provide a very large amount of 'stirring'? These planets are also much more massive than Europa which is only 0.008 times the mass of the Earth compared to ~.6 and ~1.3 Earth masses for the ones in the habitable zone. They may have a more substantial mantle and core, possibly global magnetic fields and definitely more gravity to hold on to any fugitive gasses.

Considering geological timescales, this system is also billions of years younger than our own, potentially only a little over 500 million years old. The processes you draw concern to may simply not have had time to play out yet.

I'm also not convinced that the atmosphere would be cold enough to freeze out gasses on the night side if there were significant oceans or enough atmospheric circulation to transfer heat from the day side.