It’s a question that I have heard hundreds, if not thousands, of times over the last 14-odd years working in digital marketing agencies that I have helped start, build and manage.
Every time I met with a client to go over their website and marketing strategy it would always be part of the conversation. The answer is actually not that complicated and the main point to take into consideration is this:
What do you want to achieve with your website?
Knowing this helps determine what direction you should go in when you're talking about Search Engine Optimisation.
If you're getting a new website to replace an existing older site then yes, you need a chunk of SEO to ensure any gains your existing site has are not lost and whichever company you're working with needs to make sure that when your new site goes live that it's done smoothly and with the smallest impact on your long term gains.
If you're getting your first website for a new business then much depends on what you what the new website to do.
If all you need is a brochure site to showcase what you do so that you can send prospects to it so they can see you're great at what you do then you only need to get the optimisation basics in place and let your site grow organically over time - this one is never about aggressively growing your website to drive sales.
But remember to add a blog and write about what you do and love in your business. Make it 400+words add some lovely images and make sure that it is on-point.
If you need a website that sells what you offer and you want traffic coming to the site so that you can drive in business and get the leads happening then you need The Works. You'll need strong content creation and keyword research to ensure that your website is full of the best performing keywords. You'll also need a healthy dose of all the ingredients that Google loves and then you'll need to get busy with a weekly, 400+ word blog, strong and important industry relevant website links, a well setup and fully optimised Google My Business listing with weekly articles being published on that platform (they only last a week). You'll also need a strategy on gaining Facebook and Google My Business reviews as these will certainly add to the strength of your listing.
My advice to close off is that you find someone that you know will do a good job. No smoke and mirrors. Get reviews, speak to their clients and see how they've delivered in the past. There are hundreds of SEO “experts” out there - it’s vital that you get one that knows what they are doing.
Hope that helps.