Thank you Martinn and Tooki, much appreciated!! 
I updated the original post with the additional details you asked, copied here too:
2) related charger IC:
a) suitable for that battery specific pack/chemistry
b) standalone and as simple as possible. No need to have any microcontroller control.
c) Only a mean for the Micro to read the battery level, but even that not that important.
d) the input (only when charging the battery) is a wall adaptor and I can choose any voltage accordingly to whatever is required by the IC/battery. It does not need to be a USB charger NOR a 5V charger. Whatever input voltage the IC accepts I can find a wall-adaptor to work with it.
e) I guess for safety it might need some monitoring and cutoff perhaps?
f) OPTIONAL (would be nice but no need to have) is means for the charger IC to allow to also supply the system (i.e. feeding whatever the incoming DC voltage is to my onboard switching regulators)
Well, those features take you a bit beyond a super basic charger, actually!
a) Given
b) OK
c) Battery charger ICs don't actually do that. As it turns out, accurate battery charge level calculation is
hard, so it's the purview of dedicated "fuel gauge" ICs. The vast majority of these are designed for big-development-budget large-volume gadgets whose engineers can devote a lot of time to carefully characterizing both the battery and the load to produce a complex (~200 parameters, IIRC!) profile that's uploaded into the fuel gauge. A small selection are "easy" models that just require a few basic parameters (capacity, voltage ranges, etc.), in exchange for reduced precision. Also, most of these are little tiny BGAs poorly suited to manual assembly.
d) OK
e) The whole point of a battery charger IC is to charge a battery correctly, requiring it to perform monitoring and act accordingly, starting, stopping, or changing charging as needed. On the other hand, if you're talking about pure safety cutoffs (severe undervoltage, overvoltage, or short-circuit), that's the responsibility of a battery protection IC, another discrete device.
f) TI calls that "power path management" or something. It's a common feature.
Take a look at the solution I decided on for our gadget. It's essentially borrowed from TI's
TIDA-01182 reference design, copying U1 (BQ25895 charger IC) and U5 (protection IC) directly, but replacing U4 (fuel gauge) with the BQ27441-G1, which is one of the "easy" chips in a somewhat less hateful SON package, and for which I found an Arduino library to make coding easier. We'll be using a single-cell (well, multiple in parallel) 18650 Li-Ion, which gets boosted to a 4.5V system voltage which then feeds various point-of-load DC-DC converters to produce 1.8V, 3.3V, ±5V, 7.5V, and dynamic 6-18V.
The BQ29895 gives us the flexibility to use any input from 3.9-14V (it's optimized for 9-12), including USB, will power the load from the DC supply but augment it with the battery if needed (so you can use a wimpier AC adapter), and it lets you program its own minimum output voltage from 3.5-4.5V, so that you don't need your downstream DC-DC converters to support wide input voltages. (Simpler chargers have the load in parallel with the battery, meaning the load must be able to work from both the battery's minimum voltage (typically 2.5-2.9V for a single Li-Ion) all the way up to the battery's maximum voltage while charging (typically 4.2V for a single Li-Ion). The "NVDC" system used here means that the output will go to the higher of a. the minimum system voltage you program and b. the current battery voltage, eliminating all the really low voltages.) It's fully MCU-controllable, but also supports resistor dividers to configure current limits for standalone operation. It also supports a "ship mode" where the battery is practically completely disconnected to prevent deep discharging for storage, obviating the need for a physical cutoff switch like in treez' design. (One can easily use this feature for an essentially "true power off".)
If your load is small enough, you could just use the 3.5V system output voltage and a small LDO linear regulator to drop it down to 3.3V.
I just received the PCBs on Thursday and the parts will arrive Monday, I'll let you know how it works out!