I started cleaning this up a little but I think there is no point, I'm going to get this PCB made and then change the design to use one of the TI IC's with integrated boost. That way I can have boost plus charging all in one IC.
Uh careful - those BQxxx chips only provide a USB OTG boost converter function (aka if you activate the function, you will get 5V/1A out of the normally input Vbus pin to power any device that could be connected to e.g. a smartphone). Otherwise they work as a sort-of buck converter
when you have charger connected. If the battery is too discharged or not present the IC will regulate the output voltage to about 3.5V minimum, otherwise it keeps the output at battery voltage + 150mV (or so) - so the output is not really regulated and moves as the battery voltage changes. When running without the charger, the output
isn't regulated at all, i.e. no buck or boost functionality (ignoring the OTG thing).
The datasheet is a very confused mess with quite a few contradictory parts but when you look at the block diagram of the IC it is clear what it can actually do.
If you need regulated output (boost converter), you will need to either use the OTG function (may not be practical) or add a separate converter. Basically these chips are meant to be fast USB chargers supporting OTG power and managing the power path in such way that it balances both the charging of the battery and powering the output in various charger/battery combinations. Nothing more, really.
That is unfortunate but it kinda makes sense given these are meant for smartphones and similar applications, where you will need to generate multiple power rails anyway. So it doesn't try to be smart and get in the way (which would only waste power) - it only handles the battery charging for you and delivers a meaningful (not necessarily stable or regulated) voltage that you can then convert further, as needed.