The voltage on SYS is not stabilized. It will track the battery voltage. Hence you have to add a downstream regulator in front of your system. Since you want 5 V you need a buck converter that boosts 3.7 V up to 5 V (at the cost of ~1.5 A including conversion losses (if your battery supports a discharge current of 5 A) leaving ~3.4 A @ 5 V for system supply.
Efficiency of buck converters is good, but this will add more parts to the BOM and costs PCB space.
If you can let go the USB PD/fast charging support and maybe focus on USB A legacy support (500 mA or USB 3.x @900 mA or a total of 4.5 W) you can find a PMIC that integrates a charger (e.g. Li-Ion/Li-Po) and buck converters to provide your system with one or more stable power rails including power path management.
Charging would be slower, but given you space constraints fast charging appears to be not feasible anyways.
Maybe you can stack PCBs to add more space?
May I ask what you need the INA220 for?
The chargers usually come with full current management any sometimes also with an charge/input current sensing interface. Maybe you could drop the INA220 from your application because I think you will have a hard time to find a device that integrates everything with your required specifications (e.g. low RDS(on) for the power MOSFET, >=5 A, >=20 V, battery chemistry, battery cell count, sink/source role, special features etc.). Especially for that high currents device usually don’t integrate USB PD controllers to improve thermal management.
I highly recommend to find a USB PD capable charger and pair it with a dedicated USB PD controller. Then check their package and look for the smallest. They usually only add e.g. 4 mm × 4 mm (QFN-24) + minimal discrete circuitry or even smaller.
For example, checkout the MPQ5031-AEC1 | Single-Port USB PD Controller for Source Only, AEC-Q100 Qualified | MPS (source-only) to get an idea of the full feature set of a modern USB PD controller. Then you understand better why you won’t find a fully integrated solution for your advanced requirements. The MPF52002 is available in a QFN-24 (4mmx4mm) package.
You can nicely pair the MPF52002 with e.g. MP2770 or MP2760 chargers (or whatever USB PDB ready charger you like). I recommend a charger with external power FET for better thermal management and full control over MOSFET efficiency (you can pick best RDS(on) or FOM).
Also if you want to go full 5 A maybe at 20 V (100 W in OTG mode and ~22 W charging) you should use a multilayer PCB for thermal management anyway (e.g. 4 layers for a GND plane you can also use as heatsink). If you are that tight on space you can’t omit a multilayer PCB if you want to support high power charging or sourcing.
You need heat sinking and wide power traces that will consume additional space. A typical 2 layer PCB supports ~2 A. Then you have to add much more copper (e.g. 2+ oz thick traces). Going 4 layers will be significantly cheaper.
I’m just saying in case you are not already designing on 4 PCB layers.
The MP2760 even supports temperature monitoring for the power MOSFET or alternatively the USB Type-C connector (which you should definitely consider to do - both). This way you don’t have to implement a custom heat management algorithm that includes the MOSFET/USB C connector NTC as input.
No current sensing required. Maybe you can drop it to get back some space?