Hi Sarah,
Welcome to the forum,
More often than not I see web designers claiming the complete opposite and encouraging clients to do such things, seems you've got your self a good web designer
Footer links
As a general rule footer links can be seen in poor light, if they are real navigational links then it could ask questions as to why they are needed outside of the real navigation but "sometimes" they can still be useful to users, that would depend on the site in question and its navigational usage.
If they are keyword related phrases with the sole purpose ranking for those search terms, then yes its spammy and doing so could do more harm than good.
Possible thoughts
What things could you do to help your cause, there are things out of the box which should ( in my opinion ) be out of the box with any new website. Some are painfully obvious, others may not.
1. Mobile friendly
2. Mobile page speed as good or very close to desktop page speed. ( check here :
https://developers.google.com/speed/pagespeed/insights/ )
3. Page speed in general is important, so compressing images as mentioned in the above link will help massively
4. Server speed / response times, should be good enough to cater for even the slower internet connections :
http://webpagetest.org/
5. https / SSL certificate ( green padlock )
6. Correct usage of h1, h2, h3 tags within pages and content.
7. Schema / JSON-LD Markup where possible, whether it be product, service, local business and organisational schema. It will help, depending on your web designers knowledge this maybe beyond normal remit of a web design package, incorrect implementation could have negative effects.
8. Optimise page titles, meta descriptions where possible. If using wordpress there are various plugins to help with this, just be aware of adding plugins can cause bloat and additional load times and so pick carefully.
9. Ensure correct redirects are in place, such as www or non-www , http - https etc
10. No index any pages that don't need to be indexed or could be considered to be thin content ( often pages such as categories, tags etc. Decide what you want indexed prior to going live.
11. Sitemap, both user sitemap and XML sitemap. Only include pages in the sitemap that are going to be indexed.
I've probably missed one or two things but that should cater for the basics
Good luck with your new site launch.