Given that you can't easily use google apps for free any more, here's the solution I use to get the gmail experience with my own domain(s) and without setting up any intermediate server:
Most domain registrars offer a free email forwarding service as part of the domain registration package. I have used several - currently I use Godaddy. Use that to forward the mail to your gmail account.
Next, set up gmail so that you can send email that appears to come from your domain (Settings --> Accounts --> Send mail as). This must be done after you've set up the forwarding so that you will receive the verification email that gmail sends out. Once you verify, you can also set that to be the default address ("make default") from which you send mail. In addition, you may find it useful to enable "When replying to a message: (*) Reply from the same address the message was sent to"
- Choose the behavior you want for "Treat as an alias" - click gmail's link to understand what fits your needs. From your question, I think you will want to leave that option checked.
to better hide the fact that gmail is involved, you can select "Send through SMTP servers", assuming that either you use your own server, or, as I do, you can use your registrar's SMTP servers (godaddy includes that for free with domain registration - up to a limit). This may not be necessary and there are many advantages to using gmail's servers. I suggest you try that first & if you don't like the behavior, you can always change it later.
If you are using godaddy, I can share with you the settings for their SMTP servers, which I use. If not, go to your registrar's support pages and get the info for sending with a generic mail client through their SMTP server - that's the info you'll need to supply to gmail, if you decide to go that route.
In reply to OP's comment: Every registrar I've used (a half-dozen over the last 15 years) offers free mail forwarding. Most (not all) offered free SMTP relays. As long as you can get the mail forwarded to your gmail account(s), you can implement the plan described above so that you use gmail, but the email appears to come from your domain. So you can start with that approach and when you need to move registrar's, just set up mail forwarding from them.
However - one serious disadvantage of this approach, that I neglected to mention earlier, is that you are at the mercy of the registrar's policies. For example, godaddy's servers have limits on the size of email sent (5MB, last time I checked) - gmail's limit is much higher. Also, godaddy has its own spam filtering rules, which I don't always "agree" with. Something to check if you want to use my approach.
If you don't mind running your own mail server, you can use that to forward incoming mail to your gmail account (or use gmail's POP pickup feature, if you want to store it on your server) & as your SMTP relay - then you set the policies for both incoming and outgoing mail, while enjoying the gmail web experience. The downside is you have to maintain your own mail server, but it does give you total control.