Is It insensitive/disrespectful to try to initiate conversations with employees at a company about your aspirations to join if there has been recent layoffs within their organization? E.g. I invite someone to a lunch and ask for advice on how to improve my prospects for a future position at this company? My aspirations to join has been quite constant during a year or so, but I don't know what's the right thing to do right now given the current circumstances.
I don't think it is insensitive. But companies that usually do layoffs end up going into a hiring freeze. Once they come out of the hiring freeze they try to be careful and attempt to fill open headcounts by allowing people to move across teams.
However based on your level that company might still be hiring, so I wouldn't call it insensitive and there is nothing wrong to be prepared for the time they'll start hiring.
It's definitely not disrespectful, in fact those folks may appreciate having someone outside of the company to vent about the layoff to. It's also important for you to get a sense of what the post-layoff culture looks like. At some companies there's very little change, others an 180 flip (e.g. Twitter 🐥), so as an prospect you would want to see if the culture is still a good fit for you.
From my experience, the best companies to work for are those that are transparent and honest. If a company recently did layoffs and they're not open to discussing their financial stability with interview candidates, that's at least a yellow flag IMHO.
That being said, you obviously want to treat the topic delicately and with respect. Use basic communication techniques like showing empathetic body language to start the conversation here.