Any pointers on how hard it might be to get my local business to page one?

[DixonHandyman]

Hello all.
I'm new here.

So a few weeks ago I launched my website which I built myself through wix.
It's for my local Handyman business, which I've been running for a while beforehand.

I am only really a novice when it comes to SEO, but I put the site together very decently, and the Moz checks all seem to agree my site is far better optimized than some of my key competitors sites in the local area, which I have extensively checked out.

However my question is:

Roughly speaking, how many backlinks (or backlinking domains) will i need to get it into the top 10 organic results on Google?
To cut a long story short, I can't get my business verified through gmb, so I have no hope of getting any Google reviews whatsoever or even paid ads. My competitors have the following stats:

#4 has 49 Google reviews

#6 has 156 Google reviews

#7 has 11

#9 & #10 have zero reviews, and practically non-existent Backlink profiles according to Moz checker.

To top it off, really none of my competitors have what I understand as a satisfactory Backlink profile, even the guy at #4 has some weird links in there when I run the check and he has 18 linking domains total.

#1-3 are the usual checkatrade etc, #5 is yell & #8 is taskrabbit.

To give some idea of my own efforts:

The site went live on 22 September, so it's been live 13 days.

I should have roughly 12 backlinks so far, all directory listings:

Facebook.com
LinkedIn.com
Yell.com
Handymanlocations.co.uk
Mylocalservices.co.uk
Cylex
Britaine
Freeindex
Centralindex
Showmelocal.com
Brownbook.net
Buckinghamshire Business First

HOWEVER only one of these is showing on moz Backlink checker. (Yell.com nofollow)

I've done a lot of reading and all the articles say "get tons of backlinks, but very few directory listings", but this seems practically impossible for the type of local Handyman business I have. None of my competitors sites have diverse Backlink profiles either.

So, so far my position is fluctuating from anywhere between #22 to #33 on Google.

My budget is fairly small, but I'm willing to pay to get my site to the top if I have too.

Can anyone shed some light as to if I can get there myself, or will I need help?

Please help!
Thank you.