Investors should be opening doors for you without the condition of additional equity. The value they can bring to you should be why you choose to go with them versus others. If you have other investors interested who are willing to provide value without strange conditions, these investors will be more of a burden than helpful.
Either they invest at the terms and are helpful, or they say that the terms don't work and you change the terms that they invest in and they are helpful. It sounds like they want to get in and then change the terms by doing the thing that they should naturally be doing or you could just take the money and expect no help from them.
This is sleazy and predatory, and is a huge red flag. It's not uncommon for investors to put a ton of bells and whistles into their investment (especially tied to milestones), but wanting equity purely for introductions (whether or not those introductions lead anywhere) is poor form. Don't waste time with people who act this way.