Listing syndication is done via an IDX. You can search for this term and find lots of discussion about the matter, much of it controversial. The source of the listings is through the local or regional MLS. This is a highly competitive market with well capitalized models competing for buyer attention.
I have advised numerous real estate tech companies and created them as well. $5k per month is the data cost. The mls does not allow any of these type of companies direct access to their data. The aggregated data from sources is typically old and not highly accurate. The way their data is current comes when a realtor signs up and imports their idx feed to the site. How critically dependent is your concept on mls listings data?