I'm Aria, a Social Care guru at Shopify.
You're right in saying, reputable credit card gateways such as Stripe and Shopify Payments require a valid telephone number to set it up. This number will appear on your customer's invoice. If you're not comfortable putting your personal number on there, you would need to get another number somehow.
I understand where you're coming from. I am also a solopreneur with my own Shopify store and I had to provide my phone number for my gateway. I was apprehensive at first but nobody has ever called me. They will know at a glance what the charge is for on their statement because I've got a very distinctive store name that they won't forget. I also have clear contact details on my website. That would be my advice to you - make it really quick and easy for your customers to get in touch using their chosen communication method. Make sure you respond quickly to every message. This way, they won't go looking for a phone number. I would consider the following essential:
A Contact Us page with an email form (so the customer doesn't need to open their email account to send the email).
A physical mailing address at the top of the Contact Us page.
The Facebook Messenger Channel. It's great for up-selling as well. I've found all of my customers have chosen to use Social Media to reach out to me, none of them has emailed me yet or used the mailing address I provided.
Social Media pages well maintained and linked to your footer (Online Store > Themes > Customize > Theme Settings > Social Media). I would choose two platforms. Depending on what you are selling, Facebook and Instagram are usually a good bet.
If you do all of this, it will help build trust with your customers and make your store look more professional. Customers don't like shopping with an anonymous store.
Does this make sense?
Thanks a lot for your very comprehensive response ! It sure does make sense and I didn't think from this point of view. Plus, I didn't find any other experience mentioning how they've dealt with it so I appreciate your feedback. I think I'll follow your advice, thanks again for taking some time to share it !