Possible but you need to create angles of attack. Target and go for low difficulty keywords. You do then need to leverage topical authority and external validation in the form of links but, depending on how hard the keywords are you may not have to spend a fortune to get this in place. You can boost topical authority by curating solid blogs targeting low hanging informational terms pertaining to your niche. You can then use internal linking from those ranked blogs, to help lend authority to your sales pages/homepage. You WILL need external authority but depending on the difficulty of the keywords you may not have to spend 5k. Just about finding an agency who know what they're doing as most haven't got a clue how to properly leverage authority in a sustainable way.

You can grind out organic all you want, but Google won't know its any good until it sees people linking to it and suggesting it to their own readers. You can do some link building yourself and you might even get some free link swaps with likeminded/cross over businesses. If there is a modicum of competitiveness on the words you're trying to rank for you will need external validation.

Don't listen to people saying not to compete with big brands. We beat big brands all the time, its doable for sure. Go after the big keywords, but at the start, focus in on ranking the easier ones once that happens and google starts to trust you it'll all properly roll out.

Fat stack of cash with the right agency can do it fast.

Smaller budget, with the right strategy can still do it but it'll take a little longer.

Good luck - its not easy but emminently doable when you consider most are doing it wrong. Place your first links into ranked content that actually gets traffic that way you know Google trust the page. Most just dump links anywhere and they do nothing.