Links without keyword anchor text?

Sounds like that SEO firm doesn't know what they are talking about. Anchor text is still a ranking signal. Google has gotten better at understanding the content around a link and making a judgement on what the link is likely about, but anchor text still makes a difference.

No offense, but no you don't have have good onsite SEO or good backlinks. Well, not good enough for the keywords you're targeting.

So here's the question, are you being realistic with your SERP competition compared to your own SEO skills? What I mean is, are you targeting tough keywords?

To answer your question, pages can be ranked without anchor-text. I would pick anchor-text links over plain URL links but not having anchor-text links isn't a deal breaker for ranking pages on Google SERPs.

I don't what you have heard earlier but good relevant anchors texts are still a ranking factor. For high competitive keywords they are always effective.

anchors are great but do not overdo

As others have stated, anchor text is still a signal but it isn't a make or break one. If you had the right links, from the right pages of the right sites, and were going after a keyword phrase that you stood any chance of ranking well for, the anchor text may help it nudge up slightly but I think it is far more likely that the sites linking to you are not good and/or you are trying to rank for something unrealistic.

Links for Anchor text gives a great response to Ranking signal. when we target the low competitive keyword we use anchor text and increase ranking.

If your going out building links then your treading a really dangerous path, if google detect a pattern in your activities then they will issue a penalty simple. In using keyword rich anchor text makes it really easy for them to spot as well so that could be why your SEO guys are warning you off. Normally the text "click here" or your brand name is safer but still dangerous if over done. The other issue is nowadays most links tend to be no follow links so if that's what you have been building then that will have zero effect on your ranking. Nofollow links are great for marketing your company so can bring you in traffic but in terms of helping you rank I am afraid they dont work.

Relevance from backlinks does help but hard to qualify but generally say you sell flowers and you get a flower grower to link to you then that would carry a bit more weight than say a construction web site. How relevant the source site is depends a lot on google's AI so its not really possible to guide you on that but you hopefully would know what niches are close to yours.