It is certainly possible to get along with a two channel scope (and 2 probes of equivalent bandwidth), it was done that way for decades. It's a fairly recent development that 4 channels isn't a lot more expensive than 2 channels.
Given a 2 channel scope, you can make the case for a logic analyzer and, in fact, if you had more than 4 signals to watch, it wouldn't matter which scope you had, if it didn't include a logic analyzer, you would have to buy one extra.
A good logic analyzer of some kind is quite helpful in digital electronics whether you have a scope or not.
Your approach of using a high bandwidth 2 channel scope to check signal integrity (and timing) and following up with a logic analyzer is correct.
I used a 2 channel scope off and on for over 50 years and I still have a 350 MHz Tektronix 485 for the high speed stuff. My 4 channel Rigol DS1054Z was a nice addition. Today, I would be looking at the 200 MHz Siglent offerings. I would probably still go for 4 channels. More is always better.