I'm not sure if unlimited short rests are a good idea. As someone who loves playing warlocks and multiclassing in DnD, I can assure you there are crazy shenanigans you can do if there is no limit on the amount of short rests you can take between long rests.
In my opinion the best way to discourage long rests would be to make it so that some things in the world happen based on the amount of long rests you took while still only allowing a fixed amout of short rests per long rest. In my opinion that would even increase the immersion. RIght now everyone tells me to hurry, but if I spend 10 days in a row only sleeping and eating, nothing really happens.
You could also adjust the game difficulty that way. On higher difficulties you'd have to do more encounters per day if you want to complete them all.
I'd rather not. I like to play at my own pace and hate being rushed by games. What's worse, such a system causes the game to force you to play it in a specific order.
The current system works fine. You can rest as long as you want and if you don't reach the burning inn, it will burn forever, but if you get to the location, you have a limited time to do something. Of course, main quests shouldn't have any time limit. Of course, main quests shouldn't have any time limit.