Visit your new home objectives not being completed

No, they will not. The in-game help guide even mentions this:

After purchasing a house, you can safely leave your items inside — They will always be there when you return.

Homes provide many benefits:

• Organization: There are lots of storage spaces that allow you to sort all the junk you tend to collect over time into manageable groups. Scattering all of your junk on the floor makes it harder to pick through later on. For example, I have a chest where I stash all my smithing stuff (ores, bones, leather, pelts, etc). Another place where I place all my alchemy ingredients, and another for all my soul gems, etc.
• Convenience: In most homes (other than Whiterun), there is an alchemy table next to an enchanting table. Some even have cooking areas, tanning stations, etc. This makes it much more efficient when you take into the fact that you have access to nearby storage containers.
• Spouse Management: Having a home will also allow you to send your spouse (should you choose to get married) to live there rather than having to find them or travel to an out of the way location. This is really handy for certain spouses who do not have a home and travels around a lot.
• Vanity: Every home have some sort of way for you to display items. From book shelves to weapon racks, weapon plaques, dagger display cases, mannequins, etc. Depending on how much of this you care about, being able to walk into a home stocked up on stuff like this gives you a great sense of accomplishment. You can also do silly things like drop all your gems onto a table (or into a bowl) and literally watch your treasure pile up.
• Social Status: Some holds require you to own property before allowing you to become a Thane, which grants you a housecarl for a follower as well as the ability to make guards look the other way for crimes.