I looked at this

https://www.analog.com/en/design-center/ltpowercad.html

and this

https://webench.ti.com/power-designer/switching-regulator

But nothing seems to support battery "power path" backup. In fact, from what I can see so far there's no option for battery other than a source to begin with. Am I doing something wrong?

 

Yes .........
your making it much more complicated than it needs to be.

What You need is a Voltage-Regulated, Current-Limited, Power-Supply
that is capable of both running your project,
and Charging the Battery at an acceptable rate, at the same time.
The very first thing to do is to determine the
Maximum, and Minimum, Voltage-Range,
and the Minimum-Current-requirements,
that will insure reliable operation of your project.

First, set some priorities, and some "must-haves", and "can't-haves", for your project.
Write them down, they can, of course, be changed later.

Second, select a Battery-type, and Battery-size, that fits into the above criteria,
with the lowest practical Voltage-Rating that will
still run your project, even when the Battery starts becoming Dis-Charged.

Third, determine how much time is acceptable for a complete, 0 to 100%, Charging-Cycle.

Fourth, select a Charger-type that is compatible with the Battery-type chosen,
and insure that it will meet the desired Charge-Time requirements.
( Charge-Time requirements will be affected by the Battery-Type chosen )

Fifth, decide whether You want to build it yourself, or buy pre-made stuff that will do the job.
.
.
.

 

Yes .........
your making it much more complicated than it needs to be.

What You need is a Voltage-Regulated, Current-Limited, Power-Supply
that is capable of both running your project,
and Charging the Battery at an acceptable rate, at the same time.
The very first thing to do is to determine the
Maximum, and Minimum, Voltage-Range,
and the Minimum-Current-requirements,
that will insure reliable operation of your project.

First, set some priorities, and some "must-haves", and "can't-haves", for your project.
Write them down, they can, of course, be changed later.

Second, select a Battery-type, and Battery-size, that fits into the above criteria,
with the lowest practical Voltage-Rating that will
still run your project, even when the Battery starts becoming Dis-Charged.

Third, determine how much time is acceptable for a complete, 0 to 100%, Charging-Cycle.

Fourth, select a Charger-type that is compatible with the Battery-type chosen,
and insure that it will meet the desired Charge-Time requirements.
( Charge-Time requirements will be affected by the Battery-Type chosen )

Fifth, decide whether You want to build it yourself, or buy pre-made stuff that will do the job.
.
.
.

OK, if so, without an LTC4412 how can I supply steady power to load if source is disconnected?

 

You would be using a Specially-Designed "Battery-Charger"
to Power the Boards, AND, Charge the Battery at the same time.

The Battery would never be disconnected from the Load,
and,
the Charger would never be disconnected from the Battery.

This is why the Operating-Voltage-Range is an important piece of information.

There is probably a built-in Voltage-Regulator on the Main-Board,
and it will have a specified Range of acceptable Voltage levels.

So far, nothing is known about the Voltage-Regulator(s) for the on-board USB-Ports.

Did your Board come with the USB-Ports pre-Installed ?, or did You plan to add them ?

What is the Voltage-Input-Range for your separate USB-Hub ?
It very likely has a Voltage-Regulator of some sort, and the Specifications will need to be known.
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