The thing is, the hope replenishment is about triple what it was in 1E¹ without songs². The "only spend hope if you know it will matter" of 1E is a great bit of making the hope stretch...
In 2E, it being as powerful per point would probably be too much. but a die or two after rolling would, to me, feel about right.
The spend-before systems I've run don't tie it to the same resource that is used to represent sanity.... for, with how hope works in both editions, when your hope gets too low you can have a psychotic break. (Not as easily in 2E, but still.
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¹: 1E: Fellowship pool just a hair over 1 per player, plus 1 for FF unharmed. Hobbits pushing that up. 2E, it's Breelanders instead of hobbits (not an improvement, IMO), and you get Heart per Fellowship Phase.
²: As for songs... each song was good for 1 fellowship point if 1 SL per member of the fellowship per Adventuring Phase, at least if song was shared in the party, provided when used, someone wasn't at full hope.
I got to pretty different relative hope recovery paces, only marginally higher in 2e than 1e. Pasting an old post below.
I get to triple in 2e v 1e before putting the two on a like for like basis, as 2e's fellowship pool only refreshes per fellowship phase in the Alpha. Once you adjust for this 2e's hope recovery is c20% higher. However, it looks like this may well change with the next 2e revision potentially moving to more frequent fellowship pool refreshes.
Naturally subject to changes in the assumptions below, and your point on Songs in 1e is worth bearing in mind too.
"Cross-posting some calculations I did on 1e v 2e hope recovery in another thread here, prompted by some helpful points from @gyrovague.
1e - 1 hope recovered (fellowship pool), +1 if fellowship focus not injured = 2 hope recovered per session, assuming focus not wounded.
2e - 1 hope recovered (fellowship pool) +0.5 (Patron's fellowship bonus, assumes 4 members of fellowship) + 4.67 (Heart rating of hope recovered per fellowship phase, characters have 14 stat points/3) = 6.167, however, this is per adventuring phase (the frequency with which the fellowship pool now refreshes) so we divide by 2.5 (as adventuring phase assumed to last 2-3 sessions according to p2 of the Alpha rules) = 2.468 hope recovered per session.
23% higher in 2e, assuming no hope gained from journeys and that adventuring phase lasts 2.5 sessions. And that 1e fellowship focus is never wounded. A material gap, but not as large as I'd have guessed. Obviously shifts further in 2e's favour if adventuring phases only take two sessions, and can slightly increase further due to hope from journeys, or if the Strengthen Fellowship undertaking is used. Thanks for prompting me to do the very thing we were talking about - look at the numbers!
10% higher if you're only Hope 4."