Here is another option. Why not build the closet to enclose the sink and the air handler? The door would then swing toward the cabinets. You then have a sink and some storage area for mops and brooms. The down side is the sink is still blocking the air handler. But it would not be too difficult to remove the sink when the air handler needs to be replaced. If you are worried about enclosing the sink, then remove it and build the closet as if it were there. Then new owner can add a sink if they want it to the existing plumbing connections.

"My kids put their laundry in it and forget about it." What happens to the laundry? You bring it downstairs to do? If your kids are of age to do their own laundry, do you want them to use this laundry room? If not, what functional space do you need? A powder room? An office? A larger bedroom? A reading nook? How large is the space and what's adjoining? Pros will ask for a floor plan or pictures. Your lifestyle is what drives how you use your house.

If your needs have changed and you’re otherwise happy in your home then the house can change with you. Certainly don’t need to move just because one room isn’t working for you. You are the only person that can determine if your house, minus a formal dining room can remain competitive on the market with every other house in your neighborhood. My only con would be that dining rooms are usually given pretty prime locations in a houses footprint, and now you want to give that space to what is normally a back of house service area. My pro would be larger functional service areas will greatly impact your life because you use them often and I know I would love a big mud room, a fantastic laundry room and extra space for the kitchen/butlers pantry. I’d convert it in a hot second.

Do you need/want the powder room? Are you sure you won't miss the door to the backyard? @emilyam819 has a nice and practical approach. If it was for me my solution would be simpler- leave powder room/backdoor. Remove doors from laundry area and closet. Use front loaders, add counter above. Add a shelf about 12" above the counter. Add a hanging rod above for shorter garments. Use closet for hanging longer garments and additional shelves. This would provide the functionality I would need without a huge expense. Of course if you don't have front load machines now, that is an expense. But possibly not as much as closing off the exterior door and moving the sink and the powder room door. Good luck!