What you describe is not universal, but it is not unheard of either. Remember that France is still a country which makes heavy use of cheques - this is somewhere with some very modern banking features, and some things which feel very archaic...
You've stated that you want to send money abroad, and to a non-Euro country. Given that, you might not actually want to use your bank for sending the money! For medium to large payments, it's probably worth instead using a currency broker. (Many exist, I quite like Currency Fair, though amongst friends and family I'd say there's a half dozen we use for different reasons!). With this, you'd make a domestic (French) euros transfer to the bank account of the currency broker, use the broker to change the money, then instruct the broker to pay out to the destination account. This typically means "domestic" transfers at each end, so no need to worry about your bank in France not supporting easy international transfers. For some kinds of amounts of money / currencies / frequencies, this could save you a fair bit on fees and get you a much better rate, but by no means on all cases, so check first!
Next option - ditch your "traditional" bank, and get an online one. There are quite a few banks (mostly it seems from other European countries) who offer online-only bank accounts in France. With these, there's no branch staff, no branch network, just a bank card and online banking. (Quite possibly no cheques, it depends). There's normally low, or sometimes no fees for these. I'm told they offer very good online transfers. Downsides are there being no branch network, often no cheques, and needing to have an established French financial presence before you can open the account. You should be able to get one now, and could always get it alongside your SG account and just use it for online-y stuff. (Do a domestic transfer from SG to your online account, then from there abroad)
Final option - get a different bank. I asked various expats in the pub, and they suggested (for the city I'm in, it's not universal!) just two banks as being good for expats. I opted for the one nearest where I'm staying, BNP Paribas, who do offer online international transfers:
As with SG, some things can only be done by your in-branch advisor, so make sure you like who you'd be getting, and that they're contactable. (My advisor was willing to put up with the faff of opening an account for me with mostly English documents, and has given me her phone number and email address so I can sort some things out remotely, but I'm told not all are that good...). I'm certainly paying more for my BNP account than friends using online-only banks do for theirs, and the fees/fx rate BNP offer aren't as good as a currency broker, but things are easy (by French banking standards), there's a national network of banks where I can do some things in person, and they'd let me open an account!
So, I'd suggest you ask around amongst other expats on bank suggestions, check out the advantages and disadvantages of currency brokers, investigate if you could cope with an online-only bank, then change as needed for your situation!