Next cost up, a servo tester.  Googling for an "rc servo tester circuit" will provide lots of DIY links.  Ready made ones are available.

Simplest and most reliable is to just replace the existing electronics with new 2.4GHz gear, which is reliable and with a bit of shopping around, not too expensive.  Quite often a good deal can be had including new, guaranteed, servos. It will also be legal to use and unlikely to either cause or be susceptible to interference with other users.

Other items in the possible problem list - TX looking powered up, but not transmitting.  Lots of causes.  RX not working, could just need a new aerial wire, could be something dead inside it.  Dead crystal at either end, might be as simple as cleaning the legs/sockets.  Black wire disease in the RX battery wire.  I tend to check for this by connecting the battery and checking the current I can get out of the other end with an ammeter (maybe a bit naughty, but doing it quickly and disconnecting before things heat up really is the best check).  When powering up, do the servos twitch?  If not, either they are not getting power or they are dead,  

If its a brushed motor, does it spin up connected direct to the drive battery?

What kind of motor/speed control?  Mention of 2 servos makes me think that a resistance control or a servo operated switch is fitted.  Replacing this with an ESC could be a good idea.