1. Scrape results from other search engines, similar to Searx
2. Build a Bing wrapper, similar to DuckDuckGo, Ecosia, et al.
With Ask.Moe I've tried both. The problem with scraping other search engines is that they (especially Google) are very quick to CAPTCHA your requests, so you'd need to setup and maintain tons of proxies and you'd likely struggle with search engines returning results based on the proxy location since they like to ignore URL parameters/user settings. The problem with building a Bing wrapper is that it's incredibly expensive ($3/1000 transactions for just Web, $7/1000 if you want Web/News/Image/Video/etc. as well). Let's say reaching the frontpage of Reddit/HN netted you 100.000 visitors submitting an average of 10 queries - that would end up costing you $3000 or $7000 depending on which plan you were using. And you most likely wouldn't have any ads/affiliate links to offset the costs since Microsoft Advertising (I believe that's what DuckDuckGo/Ecosia use) is invite-only and won't even entertain talking to you about your new search engine.
The reason you see a new search engine pop-up every month and disappear again the next is because it's easy to build a Bing wrapper but incredibly difficult to make it profitable.
My plan is to build various categories (so far I've built https://ask.moe/math and https://ask.moe/currency - podcast or news will likely be the next) that are inexpensive to operate and can hopefully generate some revenue. I'm also strongly considering supporting accounts where users can buy credits to use the general-purpose search engine, but I think it will be very difficult to convince people to pay to use a search engine.