There's at least two security concepts here that you're concerned about: vulnerability and availability.
First, vulnerability. Anything that is network-connected is theoretically network-exploitable. So adding another device to your network does increase your risk. A NAT firewall at your home network's boundary would prevent incoming connections. But there are at least two more attack vectors. If someone is able to get onto your WiFi and join your network, then they can talk directly to your washing machine. Or the app you use to talk to the washing machine may itself be vulnerable. If an attacker has the tools to install malware on your washing machine then you could be contributing to a bot net, which could create a lot of malicious network traffic.
Second, availability. You specifically asked about a requirement for online access in order for the appliance to work. I can't find any situation yet where that happens. But it definitely happens with video games. So it's an option that companies are aware of. This situation is related to digital rights management (DRM). In my opinion, the bigger availability problem is if the device relies on an external server in order to function, and then the company drops support and turns off the server, making the appliance useless. Amazon has done this to a couple different devices although none of them are comparable to a home appliance like a washing machine.
A final possibility that you didn't mention was privacy. Is your appliance collecting data about you and sending that data back out? Will your appliance start showing you ads every time you want to wash something?
I do own an oven that is WiFi-enabled. It is not connected to my network.
