To answere your original question, NO a weak remote battery will not cause the jeep to be hard to start

There's a chip in the key (with or witout the remote) that reacts with the imobolizer. If that chip is not programmed for your car or is bad the jeep will start but shut off in like 2 or 3 seconds.
I bought a blank key without the buttons to use when I expect to get wet. Like when washing the jeep or are at the beach. Word or warning. If you have 2 working keys you can have a third cut and then you program it yourself

But if you only have one working key the second will be a expensive trip to the dealer. But I understand SOME, not all, lock shops can program a second key for you. My blank on flea bay was was $20 and a local shop cut it for NC since I went to him first and he couldn't do it

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