The reality is that advances in banking convenience have focused on what 99% of people need, which is services in their own currency only. I think you will be hard pressed to find any solution where you don't need to physically deposit foreign currency cheques. Cross-border and multi-currency banking (at least in North America) has evolved amazingly little from the 1950s (maybe slightly exaggerated, but riffing off your question).
You *might* find it helpful to open an RBC USD small business account. I have done so for my business, since I deal with both CAD and USD income and expenses. It costs $9/mo but waived if you maintain a balance of $2500+. However, the value of doing so comes not from saving time from going to the bank -- I don't think you can deposit USD cheques to it at an ATM, only teller (may be wrong on this). It comes from saving the foreign currency spread on conversions, and being more easily able to match your USD payables and receivables.
My personal accts are at a different bank than RBC, but I know RBC has a subsidiary in the US. If you open a USD account there, you can deposit US cheques to it electronically, just via an app. Then you can transfer between your US-domiciled and Canadian-domiciled RBC accounts. I don't know if you can do this for small business accounts, but worth asking.
I suspect you won't find an easy solution to avoid an in-person deposit of the Euro cheque, however. So therefore may or may not be worth jumping through hoops on the U.S. side to only solve half of your problem.