On the other hand the design is partly the fault of DnD in general. The "heroes" of this universe are very quickly super beings who mainly do the big stuff, sadly. The setting you describe for BG1 sounds bucolic but how can a group of nobodies become semi-gods over the small timeframe a game takes place? Which they do. That's actually nonsense, more nonsense than to be accompanied by a vampiric elf and been confronted with mind flayers at start.
Sorry to pile in but I think you're being a bit hyperbolic here. No character or companion in BG1 becomes a demi-God. Characters in BG1 reach level 8-10 if I remember correctly?
The only character in the original BG games who evolves into something eventually approaching a Demi-God is our player character but that seems mostly on account of his rather interesting background (didn't want to post spoilers here). The game doesn't not seem to imply taking place over a short time frame either.
Don't take it too literally. It's enough that you are able to cast fireball as a nobody after a few days/months. I don't blame either game story system, there are always illogical inconsistencies with leveling systems, not only in DnD or other RPG variants. Take the Total War series, it is usually total nonsense to start in the given timeframes as a small undeveloped faction, it is however necessary to have the fun experience of progressing.
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A feeling of urgency which is soon displaced by "don't worry, the tadpole doesn't affect you or your companions on the same way it does everyone else (without explaining why) so feel free to go about your merry business". Which in my humble opinion is not an example of particularly good or immersive writing. ...
That's indeed a problem. I don't like time limits and was quite worried after EA release about each rest and delay because of the tadpole. When I realised that the tadpole is not such an urgent problem, I was relieved but also baffled. However the game tries to avoid the terrible story fault f.e. Fallout 4 has (FIND your son, but no problem if you first do this and that for a very long time) by giving hints that the tadpole is not a normal tadpole. Generally I'm not a fan of such "urgent duty" stories (btw the tadpole smells like the heavy version of the collar from D:OS2 Act 1), but let's see what is in the background.