I've heard of a few PATS issues from this generation but don't recall ever seeing it start working after a 2 minute wait. Odd.
No, the transponder does not have to match, is not programmed in any way, is just an active electromagnetic coil to energize the chip in the key, and antenna receiver to read the code that chip transmits, and send to the PCM.
Note that you might be able to hook up a scan tool capable of ford specific codes and see what code is generated, especially if the PATS dash light starts to blink rapidly as it often does if PATS is preventing starting it. If you do not have the PATS dash light blinking rapidly, then I would question whether PATS is to blame.
Besides the transponder around the ignition key cylinder, there are only two other components, the chip in the key, and the PCM engine computer. The PCM has to match the other vehicle parameters such as engine and differing sensor setups that vary some from some model years to others, but otherwise a compatible PCM just needs the keys programmed in.
You lost me at "OEM start system". There is no modding that can be done or necessary, except that a custom tune firmware can be flashed to a viable PCM to eliminate pats. Garret Tuning is one of a few businesses that will do this for a fee, around $125, or web search for "Ford PATS Delete Service".
www.garretttuning.com
However I would sooner suspect the transponder and get a junkyard pull. There may be several Ford vehicles of the era with a compatible transponder, not just Explorers but I don't have a comprehensive list. However you are in a unique situation in that an '01 Sport, switched PATS types, around 01 (7/24/00 build or later)-03 is the later PATS type E instead of B. One significant difference is the type E includes starter interrupt, so if yours is cranking then you have the older type B.
Another possible difference is, I don't know which version of pats or model year, changed to making it integral to the gauge cluster assembly. It definitely wasn't with pats type B but I don't know about E.
You can use this info and compare against a PATS Job Aid chart like the following that i've attached. I don't know if that makes a difference in which transponder is needed but would think it most likely to have success getting one from a vehicle with same PATS type, or of course there may be a Ford part # on yours that you can use to do an online search.
No, the transponder does not have to match, is not programmed in any way, is just an active electromagnetic coil to energize the chip in the key, and antenna receiver to read the code that chip transmits, and send to the PCM.
Note that you might be able to hook up a scan tool capable of ford specific codes and see what code is generated, especially if the PATS dash light starts to blink rapidly as it often does if PATS is preventing starting it. If you do not have the PATS dash light blinking rapidly, then I would question whether PATS is to blame.
Besides the transponder around the ignition key cylinder, there are only two other components, the chip in the key, and the PCM engine computer. The PCM has to match the other vehicle parameters such as engine and differing sensor setups that vary some from some model years to others, but otherwise a compatible PCM just needs the keys programmed in.
You lost me at "OEM start system". There is no modding that can be done or necessary, except that a custom tune firmware can be flashed to a viable PCM to eliminate pats. Garret Tuning is one of a few businesses that will do this for a fee, around $125, or web search for "Ford PATS Delete Service".

Ford PATS Delete Service 1996 - 2004 EEC V
Service for deleting PATS from 1996 to 2005 Ford Engine Computer. PATS stands for Passive Anti-theft System or Fords security system. This service is perfect for engine swaps or if the PATS system is malfunctioning on your vehicle. This is also useful if you no longer want to have to purchase...

However I would sooner suspect the transponder and get a junkyard pull. There may be several Ford vehicles of the era with a compatible transponder, not just Explorers but I don't have a comprehensive list. However you are in a unique situation in that an '01 Sport, switched PATS types, around 01 (7/24/00 build or later)-03 is the later PATS type E instead of B. One significant difference is the type E includes starter interrupt, so if yours is cranking then you have the older type B.
Another possible difference is, I don't know which version of pats or model year, changed to making it integral to the gauge cluster assembly. It definitely wasn't with pats type B but I don't know about E.
You can use this info and compare against a PATS Job Aid chart like the following that i've attached. I don't know if that makes a difference in which transponder is needed but would think it most likely to have success getting one from a vehicle with same PATS type, or of course there may be a Ford part # on yours that you can use to do an online search.