5e is an extremely rule-light game (six numerical bonus coming from stats, modified by a single "proficiency bonus" if you can somehow claim to be proficient in the task (whether it is because of a skil, because of a tool, or because you talked the DM into buying that you having a Kundarak bank account in your background made you an expert in cracking Kundarak safe). Expertise modifies it slightly. Tacked upon that is a rule-heavy subsystem for dealing with combat.
If you have no problem with a single roll resolution system with large variability (d20) compared to the static bonus you can apply (say, from -1 to +11), you can very well do anything with this rule-light system. Sure, there may be better rule-light systems, but it's like poker and chess, you can have fun doing both, even if one is superior to the other. And saying "you can do anything in improv theater" doesn't mean that it's lame to play D&D, it means that since you can do anything even with no rules, you can all the more do anything with the simple ruleset.
In my current campaign, the average must be of less than 1 fight by evening. In the one I play in, we're at session #4, and we fought: a few kobolds, 3 dogs and... we had an action scene with insects but it wasn't really a fight, more a skill challenge (can you repair your vehicle before the swarm reach you?). So it's at most 0.75 and in truth closer to 0.5 No player died of boredom during these sessions (50+campaign in the first case and TBH, fights are becoming increasingly rare as the campaign progress).
We could switch to another system. But why, when the one we have is perfectly functional, albeit simple?
So to answer the first question, yes I feel it can (but I am not looking toward rule-heavy).
With regard to the optional rules, tools as proficiency and background as proficiency are useful to differentiate mechanically between players. In the end it doesn't matter a lot since it's always "just the prof. bonus", but players like to leverage something on their character sheet to get an advantage in a given situation.
To maximize opportunity, I remove filler combat, I try gave a semblance of authority to the PC (so they don't have to fight their way through all situations... IRL most police forces giving speed ticket don't end up in a firefight) and I play in a urban setting. If the most dangerous random encounter is a drunkard, fireball might be out of place. On the other hand, the combat systems allow for fun and interesting fight when they happen. As alluded above, we also do things like skill challenges when there is a tension needing collective effort to be resolved.