IRFZ44N is really made for around 10 V V_GS, at 3.3 V it will conduct a little bit, but conduction below 4.5 V_GS is not even specified. IRLB8721 is made 5 to 10 V V_GS operation. 3.3V at the gate would turn it on more than the IRFZ44N but it would still be rather ineffective. RDS_on would be around 50 mOhm. You would use a huge transistor capable of switching tens of amperes to switch 2 A. And because the transistor is huge, the miller charge is huge and switching speed using a microcontroller output will be rather slow, because at the currents that the outputs of the microcontroller can supply, recharging the gate will take a long time. For using such as transistor you would typically use a MOSFET driver IC or two transistors as a push-pull configuration to supply several A to quickly recharge the gate. You should use a transistor that is made for 3.3 V operation and has an even lower threshold voltage. Gate threshold voltage is the voltage where the FET starts to turn on, but a significantly higher voltage will be needed until the transistors conducts well and can be used efficiently anywhere near it is peak current handling capabilities.
IRLML2502 should be a good choice and would even fit into the same SOT-23 footprint as your BS170. It is also specified at 2.5 V gate to source voltage with 50 mOhm RDSon. At 2 A 0.1 V would drop at the transistor and 200 mW of power would dissipate at the transistor which is well within the specification. Gate to Source (miller) charge is around 1.8 nC. That means you can charge or discharge the mosfet gate using a typical 20 mA MCU GPIO in around 100ns and no external gate driving circuit will be required.