Need to repair rotted wood on exterior door frame

Multi tool, shims/wood, caulk, sand, paint the new wood to match. That is about all you can do really.

Trim rots out at the bottom as they wick up moisture.

You are on the right track but look for a fake wood trim piece to use as the replacement for the first foot or so.

ETA: poor word choices.

After you paint it, problem solved for good.

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Trim rots out at the bottom as they wick up moisture.

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Before you replace that small piece of trim with new, be sure to put a couple coats of paint on the end grain of the trim piece to help prevent moisture being wicked up from the bottom of the new trim piece.

Post some pics if you can. I dealt with this on our old door out to the deck. It was up on the 2nd level of the house, so it wasn't sitting at ground level. There was a storm door there outside of the main door, but over the years it wouldn't quite shut all the way, and in heavy rains water would get inside the storm door. We've had the deck rebuilt and covered/screened in now, so no more water getting in. But while they were there doing the deck, I had one of the guys repair the trim. He basically did what you are saying, multitool to get the old out, primed and painted the new piece, slide it into place and tack some finishing nails into it, then repainted all of the trim so it would match. Ours was a simple 1x3 or 1x4 as the trim/casing, so easy to match. If yours is anything different and you want it to all look uniform, probably easiest to rip it all out and just replace with fresh all the way around. Wood is fairly cheap and saves you the headache of trying to match looks.

definitely just pull the whole section of trim and replace the whole piece, it's easier and will look better;
if you can't find an exact match, you could have it milled or just replace all the trim with something else, just get exterior grade wood (I prefer old cypress personally) and back and end prime everything