Two amps running through a transistor?
You've practically got all your answers in the comments above. But you also brought up cost. So I thought I'd add a few points and try and wrap it up.

Transistors commonly have to handle more than 2A. They may do this for purposes roughly like a switch (turning something ON, like a solenoid) or for purposes not at all like a switch (like the speaker driver stage of an audio amplifier.) But because transistors also drop at least some voltage across them when doing this work, they also have to dissipate power, too, while achieving their primary purpose.

So, in any design you need to also keep in mind power dissipation. There are transistors designed for small signals which are generally these days packaged into tiny TO-92 packages or SOT-23-X packages. These are not designed to dissipate much power into ambient air. So you usually have to stay under a quarter watt or so with those. Less is still better. The temp at the surface of the package means that the temp inside the package is still that much higher. So you need to make sure that the actual bit of silicon deep inside doesn't exceed its ratings. The 2 amp number you give pretty much takes it out of this ball-park of small signal devices.

However, there are packages that fit squarely into that area of current. These are TO-220 packages. And for many 2A applications, they may be just fine. Again, everything is in the particulars. There are "safe operating areas" and you may have to drop large voltages across the transistor, so you still have to think and read datasheets to make sure. But as a broad rule, the TO-220 package (and similar) probably will handle many 2A situations.

There are two basic kinds of transistors: FETs and BJTs. They each have their place, still. So which of them will be better will, again, depend on circumstances. Since you mention cost, I'd say as a broad rule BJTs are cheaper than MOSFETs. I can get BJTs for less than a penny each (I buy PN2222A at about 0.4 cents each.) I can't do that anywhere with any MOSFET, such as a 2N7002, which all have a base that is much higher. However, that's not all there is. They each require drive circuitry and that also costs money as well as having different trade-offs in performance.

But a TO-220 packaged BJT can often be found for well under 40 cents. (Assuming you get it shipped for free, I guess.) They really aren't that expensive. And I'm pretty sure similar MOSFETs aren't much different in pricing, when talking about TO-220 packages.

So it's not the big deal you suggest there. Handling 2A isn't that bad and isn't that expensive, in many applications. (Of course, for some it would be prohibitively expensive. So no bright lines here, either.) In fact, it's considered pretty normal.