There are a lot of considerations to make before investing in bullion.
As you've already noted, coins are more expensive than bars largely because of the additional artistic work that goes into them. However, there are other considerations, including tax status, between the precious metal & the coin/bar options which I mention at the bottom.
DISCLAIMER: I do NOT use these services and can not make any judgement on them. I can only tell you that they exist - this is an informational post ONLY. I can only speak about the UK's Royal Mint - other mints may offer different services or have more competitive prices, I don't know. I've just seen these options whilst on the RM site.
The Royal Mint does issue bullion bars in gold, silver, and platinum. You can see those here: https://www.royalmint.com/invest/popular-bullion/all-bars/
The gold bars do work out cheaper, obviously how much cheaper depends on the volume bought. Looking at the most expensive options on Royal Mint, which is 400oz, you would save ~£19,523 if you bought the one x 400oz bar rather than 400 x 1oz gold coins even accounting for the bulk discount. However, even the bar has some premium on it because the Mint wants some profit even from that (and likely has refinement costs to consider etc).
400oz is the “standard” gold bar you're thinking of, if you were say, visiting Fort Knox - that's why it's that weight, it's not a random arbitrary number.
The Royal Mint also has a service called digi-gold which is exactly what it sounds like. It's sort of like an online bank account, but using gold instead of normal currency. You do buy actual gold, but you never receive it and the mint holds it for you in a "big safe” somewhere.
You can buy/sell as you like. I would read more into this before using the service. I'm not sure if you can only sell to them, or whether they just guarantee they will buy it if you specifically ask them. Obviously, if the latter, they charge a fee on that. They also charge an annual service fee for holding the gold for you in their “big safe”.
They may offer this service for other precious metals, I don't know.
As you noted, you can of course buy bullion bars or bullion coins from a lot of other marketplaces. Whilst you might get a better price, you can run the risk of getting fakes or clipped (I.e. less weight than they ought to be). Usually, the more trustworthy the site, the higher the average price is as the risk is lower so more people use it. I can't speculate whether there's any sites which are as trustworthy as the Royal Mint but with lower prices as it's not something I dabble in.
If UK based: Capital Gains Tax - Bars & Digital products ARE subject to Capital Gains Tax. Bullion COINS are NOT.
If UK based: VAT - Silver & Platinum products WILL have VAT charged if buying from the Royal Mint (or legitimate business), currently at 20% - You can NOT charge VAT back if you sell, unless you're a business too. Gold is VAT exempt.
Digital Products: These will charge service fees for holding your bullion.
Physical Products: Keep them in a secure location in case of theft. Consider insurance and/or any storage fees.
Physical Coins: Many high-street bullion dealers will only offer you the intrinsic metal value and will ignore any artistic value.
Royal Mint cover a range of other bullion FAQs here: https://www.royalmint.com/faqs/bullion/
If you are considering investing considerable sums of money into bullion, speak to a specialist adviser to navigate the different tax statuses of things before making any purchases. Don't believe just anyone especially on the internet!