With 28 years in the trade, the work itself clearly isn't the problem — this is a "getting found in the right order" problem, and new companies feel it worst on Google specifically. A few things I'd check, in this order:
1. Your Google Business Profile before the website. For residential HVAC, most first calls come from Maps and the local pack, not your homepage. Make sure the profile is 100% complete: primary category set to HVAC contractor, every service you actually offer listed (AC repair, furnace install, maintenance plans…), service area set to the neighborhoods you really cover, hours, and photos of your techs and trucks — real ones, not stock.
2. Reviews are your biggest structural disadvantage right now. A shop starting from zero reviews will lose the click to a competitor with 80, even if you rank right beside them. Build the ask into every finished job from day one: tech mentions it at the walkthrough, link goes out by text within the hour. It's slow for the first couple of months and then it compounds.
3. Look at the gap honestly. Search your main services from a phone in Baton Rouge and note who shows up and how many reviews they have. That tells you whether you're fighting a visibility problem or a trust problem — the fixes are different. (Visibility and competitor tracking is most of what I do at LocalCMO, and that one search usually tells you which fight you're in.)
4. One caution on the website redo: if pages are already indexed and getting any traffic, don't let the redesign change their URLs or thin out the content. A relaunch that breaks what's already working can set you back a quarter.
SEO takes months to pay off even when it's done well. In the meantime, a complete profile, a steady review habit, and asking every happy customer who else they know will carry your phone better than any single channel. What does your review count look like right now compared to the shops showing up first?