In my jurisdiction, the realtor would have to have the client sign a document indicating they received a kickback on an inspector referral if there is one. Of course, preparing documents for realtors I've never done one of those. It's more likely that the inspector works with them often and they have trust. Also it's industry practice to suggest you use your own home inspector and at most provide a name if asked for one (do you have any you recommend? "I've before worked with xyz and it worked out").

Open houses are 99% the client expectation. Nobody wants to work an open house. Usually open houses are only to placate anxious sellers, as they view it as "part of the job". I've never known a realtor to push an open house, and often they end up paying junior agents to cover them since they don't accomplish anything except a 1% chance the junior agent could end up with a buyer client.

Also, when buying, you don't pay a dollar in real estate fees, so you saved $0 by doing it yourself. Congrats. Seller pays both realtors, when they use one. If you want to sell a house without a realtor, have at it. If you can market it yourself and make sure you have all your ducks in a row when it comes time to sign the offer, not miss any latent defect disclosures and meet all the other requirements, then great!