I have played most build combinations in BPM. With a level cap of 33 I think many can eventually do what you're talking about. The hard part is "take on any encounter at any point" as many builds come online late.
I've been playing a lot of builds in ultimate-like conditions, some with deadly deadfire on, level cap 20, testing against megabosses, DLC content and such. These builds I am confident can solo everything without cheese
FF / soul blade
SC FF
Barbarian / Devoted or Barbarian / Black Jacket
Barbarian / Ghost Heart
The last two I've only tested high level content and some of their survival abilities come on late, particularly with the savage being dependent on savage defiance for healing, so I'd run some kind of a forbidden fist or the brute.
FF / soul blade will be stupidly strong by level 33 since you'll get the high level FF abilities plus cipher's shared nightmare. SC FF already has enough penetration at L20 to beat everything, if you kept leveling it you'd just benefit from more accuracy and damage from transcendent suffering power level mainly, but that also levels with multiclass.
There are a lot of builds that can do like 90% of the content or even 99%, but many have trouble early on or can't beat specific megabosses without cheese or at least consumables. Hauani O Whe is probably impossible without a cipher or some kind of summons (without consumables, or going way out of your way to get Mechanical Marvel). Dorudugan is impossible for some tanky builds that don't do damage fast enough. I think troubadour / psion can do nearly everything by late game, I don't know about on the way, but I'm pretty sure you won't be able to beat Dorudugan.
I would advise against blood mage. It was nerfed pretty hard. Wall of Draining is much weaker and blood sacrifice prevents healing. There may still be some blood mage / X builds that can solo everything, particularly by level 33, but not as you encounter things. The strongest blood mage / X builds in BPM are mostly melee builds IMO. Troubadour / blood mage is interesting and has a lot of late-game versatility, but without consumables your healing is going to be extremely limited early on. Even once you get ancient memory plus blood mage passive you will heal something like what a fighter would get from constant recovery alone. I think this build will not be able to beat Dorudugan at L20, and HOW would be extremely difficult. But at L33, I have no idea, with the power level boosts to healing maybe it makes a big difference.
Forbidden fist builds are going to be your best bet by far IMO. In addition to the two I listed, ravagers, cantors, votaries, brawlers, wanderers, and sages (melee focused) are all quite strong, though I haven't tested them as much, and HOW while possible with dichotomous souls can be rather difficult.
With FF you get your strongest attack at Level 1, and as long as you balance INT/RES/DEX correctly and choose items that reduce hostile effect duration, you can spam that attack, which had tremendous penetration, good accuracy and damage, enfeebles enemies, heals you, and produces wounds. Pure FF fighter builds are not that interesting to me since you mostly just spam FF, part of why I like the transcendent more.
------
I should also note when you reach Tier 9 in multiclass builds due to the Level 33 mod, many multiclass builds become absurdly strong. Many classes have Tier 8 and 9 abilities that provide resource regeneration, which is reasonably balanced with a level 20 cap that limits in most cases said resource regeneration to single classes. Fighters, paladins, monks, rangers, rogues, and barbarians all gain varying amounts of resource regeneration, and certain combinations will become quite broken. Also, I'm unsure if shared nightmare works on other classes spells. If it does, then cipher/casters could become extremely overpowered when their spells cover the entire screen (e.g., a hierophant could possibly use citzal's spirit lance to hit everyone on the screen and use soul annihilation on all of them). In other cases, like paladin / troubadour, you can easily regain paladin resources from chanting many lives pass by, and fighter / x martials as well as ranger / x martials become much stronger for their resource regeneration. Until that point you would be reliant on potions of enlightenment in long battles with most classes, but you don't want to use consumables, so... this basically just leaves combinations with renewable resources, i.e., monks, chanters, ciphers, blood mages, paladins, and their various combinations. Again, FF / soul blade ticks all the boxes...