Should I Remove Chimney and Leave Firebox with Gas Logs?

We purchased a 1980's house a little over a year ago with a wood-burning fireplace. The brick chimney is on the outside of the 2-story cedar-sided house and extends through the roof and guttering. The chimney is on the back of the house inside of a fenced in yard. We have not used the fireplace because it leaks and frequently has water standing in the firebox.

When we purchased the house last year, the home inspector said that the water in the firebox was because the gutters, which butt up against either side of the chimney needed diverters to direct the water away from the chimney. The seller added diverters to the guttering, but the water leaks continued.

I had the fireplace cleaned and inspected recently by both a chimney sweep company and a contractor. The chimney is leaking in multiple places and needs repair and sealing including the top of the fireplace. The flashing also needs repair/replacement. All of the siding and trim right around the chimney is rotted and will have to be replaced. Some of the sheathing on the house is also rotted. The chimney inspection also revealed a hole in the interior of the flue which means I can't use the fireplace unless the flue is re-lined.

So, it looks like I need to repair and seal the chimney, fix the flashing, fix the top of the fireplace, line the flue with a stainless steel flue, replace the sheathing, siding, and trim. The fireplace, hearth, and mantle inside of the house look fine.

My question is this... Would it be feasible to remove the chimney down to the firebox instead of repairing and sealing, cover the opening in the firebox somehow, and install ventless gas logs? I know that I would have to replace the siding where the chimney is now and also rebuild the roofline and add guttering all the way across. Would like any ideas of options besides full repair/seal/relining of fireplace and replacement of siding, including which would be more economical.

Thanks for the help.