The only character whose approval grows that high is Gale. The second character is Shadowheart, but it's not even 50% as easy as in the case of Gale. A large loss of approval causes the player to be locked into the companion story that is a large part of the game.
Besides, I don't like to be rushed, especially in a situation where rushing will end up dying quickly.
Let's say the Lae'zel quest. There is no chance that a character below level 4 has any chance to patrol without cheese, and this is just the normal difficulty level. In the full game, playing very hard (or whatever it will be called), there is no option that you will be able to beat this fight without level 5. Most likely, in the full game, you won't be able to reach level 5 before clearing most of the map on the surface.
I don't like situations when the game is punishing me because the fight went wrong because this group of half-dead goblins massacred my character in one turn and I was forced to rest once too much.
I never argued for a large loss of approval. Even a single point of approval loss would be sufficient, simply because the display message "Lae'zel disapproves" would motivate players to push on. Also, in my initial post I did suggest allowing a persuasion check to reduce/negate that approval loss. Again, simply because the fact that the message/conversation would be brought up is enough of a disincentive to frequent long resting.
If you don't like to be rushed, that's fine. But that's a separate issue from things like: "the system will be simple to soft lock," and/or "the mechanical consequence is too punitive." I'm not arguing that the game should require you to go through 10+ fights before you're able to long rest. But 2 or 3? Yeah that's fine; that's the lower limit of expected battles per long rest in 5e. And given the prevalence of scrolls, the healing from short rests, and the class abilities that recharge on short rest, it will be extremely unlikely to reach a scenario where you literally cannot continue adventuring without long resting in such a time frame. And even if you do reach that scenario, you can rest anyway with a (possible) slight approval loss! And/or lower the difficulty level if you want (another option is that rest limits are only turned on for "hardcore/realism" difficulty modes.)
This implementation could also be combined with other metrics: i.e., if the game detects total HP across the party to be too low (i.e., exactly the example you propose above) then it could add a dialogue line to Lae'zel "You make a good point Tav; a smart warrior knows when to rest for the next fight," and reduce/negate the approval loss if you force a long rest.
