2.1 system needs 3 HDMI inputs??

Moving to a new house that won’t realistically have option for surrounds in the living room and looking for advice. My current multichannel system will be moving to the basement, but I still want decent sound quality in the main living space. Initially I was interested in switching from an AVR to an integrated amp, but can’t find one with multiple HDMI inputs. The family regularly uses a cable box, Roku, and occasionally a Blu-ray player along with streaming audio. Best I can tell that leaves me 4 options and wanted feedback from the forum about the pluses and minuses of each:

1) an integrated amp with 3+ HDMI inputs and 4K pass through that I’m not sure exists... if it does exist, I’m guessing it’s costly. 

2) run extra HDMI cables in the walls to use the TV as the HDMI switcher for all sources and then another HDMI on the eARC channel back to an integrated amp...but I’m not sure the effect that running everything through the TV would have on audio quality plus hate getting into the crawl space to run 3 additional cables and enlarge the holes (previous owner left a single cable in place at my request from the AVR they had)

3) a cheap AVR with R & L preouts connected to a better amp than the AVR would have built in...gets me solid amp but AVR quality preamp (and cheap AVR at that initially).

4) a separate box HDMI switcher (3+ in, 2 out) that works with our harmony hub and connected to an integrated amp... again not sure how good these are since I’ve never looked into them before or if lip sync issues would be possible to avoid.

Your suggestions, experiences, and advice are much appreciated. Thanks in advance. 
I have a Bryston DAC3, which has 4 HDMI inputs.  It is a great DAC, so I would get a “traditional “ integrated amp (no included DAC) and then you are set.
The only thing made that is similar to what you are looking for was the Classe Sigma 2200i integrated amp.  It has 4 HDMI 4K inputs (only one supports HDCP).  However, the audio over HDMI only supports 2-channel PCM, so you will need to configure your sources to send only 2-channel PCM.  This would be a used item.  Nobody has it up for sale now though.

Otherwise, you'll need to do an AVR or a HT Processor with a separate amp.
There are tons of devices that do what you want but most aren't cheap because if you want cheap, just buy a receiver. Arcam has (or had) a 2 channel receiver with HDMI inputs. The Lyngdorf TDAI-3400 and NAD M33 both have optional HDMI boards. The Lyngdorf TDAI-1120, NAD M10, and Arcam SA30 all have single ARC HDMI inputs if you want to run your sources to the TV and send ARC back to the integrated amp.

All of the above also have room correction built in, which might help your setup since this isn't presumably going to be a dedicated listening space with perfect room treatments. You can also look at the Naim Uniti Atom or the new Cambridge EVO series, both of which have single HDMI ARC inputs, but no room correction.

I have the TDAI-3400 in our living room and the TV plays through it. It sounds incredible, quite honestly, for a 2.1 setup. I also have a 5.1 media room and while yes it is better for surround sound, it's surprisingly closer than you'd think. Actually, dialogue intelligibility, which is critical, is far better on the Lyngdorf than in my media room with a dedicated center channel.
@jnehma1 I forgot about the optional MDC expansion modules on NAD products to give them more than a single HDMI input. That might be the ticket that avoids having to spend a day in the crawl space running extra cables through walls and includes options that won’t break the bank and keeps everything one-box simple for the wife and kids. Forgot to mention the TV is wall mounted and the audio components are staying hidden away for WAF reasons, which is why I didn’t immediately plan for option 2. 
I glanced at that Lyngdorf (as well as their 2170 model) and it/they look(s) awesome. How badly our contractors end up over budget on our renovations is going to determine how much I’ll have to play with for audio gear... I fear Lyngdorf money will end up spent on bathroom tile instead for the time being, but good to know it’s out there if I’m pleasantly surprised.

@auxinputI’ll keep an eye out for that Classe.

 That Bryston looks like an impressive option too...2 of the 3 reviews I found ended with the reviewer buying the review sample

good to know that option hasn’t had a noticeable effect on sound quality for you. I had done the same in a setup before but was plagued by lip sync issues...admittedly a much older TV and amp, however
The Arcam SR250 is a 2.1 channel AVR with HDMI switching. They discontinued them a while back so finding one new may be a chore but they are pretty easy to find used. It has Dirac room correction and a decent internal amp but it may be on the anemic side if you have hard to drive speakers. I'm using one currently in the same situation you are describing and I really like it. I have mine paired with an external amp and it works very well for my use case. 

You could also use your TV as the HDMI switcher and run an optical line back to any integrated with an internal DAC. I still do this so I can use the smart apps on my TV and run them through the main system.

The benefit with the Arcam is it can decode and downmix all the surround formats where a standard integrated won't be able to. You will have to change the output settings on the TV to send a digital signal to the DAC in a standard integrated amplifier that it can decode but it will still work.