I did a tile counter years ago in my starter home for the same reasons. I ended up using an L-shaped wood trim piece as the edging.
There’s also the retro metal counter banding and countertop edge molding. Other than wood or metal, I can’t think of a way to do it.
I’d really consider alternatives to tiling. Laminate is low cost and very functional, and while it may not be your ideal, it’s so easy to pull out when you’re ready to move on to better counters.
Since your area is so so small, you may be able to find a surprisingly affordable stone remnant or piece cut with an error (an error that’s irrelevant to you) at a stone yard. You might get lucky at a construction resale shop too (we have ReStore here).

Daltile and American Olean make a big range of trim tiles in many colors including several whites. You might try matching the sink trim tiles to the sink color instead of the porcelain tiles. Or use a laminate countertop and put the porcelain tiles on the backsplash.

https://americanolean.com/trim.cfm?tc=84

It looks like you will be upgrading your electrical to meet current code. If you want other comments on a budget kitchen redo, you can start a new post. It gets attention if you give a budget in the title, such as "1920's kitchen redo for under $1000" ($10,000, whatever).

Just a bad idea. Go get butcher block for the whole thing. The amount you are saving is not worth the saving for a lousy result. Even birch at Home Depot would be better. 50 inches by 25 x 1.5 116.00

https://www.homedepot.com/p/50-in-x-25-in-x-1-5-in-Wood-Butcher-Block-Countertop-in-Unfinished-Birch-BBCT1502550/300688696?gclid=CjwKCAjw-8nbBRBnEiwAqWt1zWuwVDtRd5nRlaKAbmIAPowd3ll2IJRiyv_Rjlv60WDwwn9vNzc0GxoClKsQAvD_BwE&dclid=CIzC2vHL7NwCFce4swodatsMhQ

Or Lumber Liq. in maple

https://www.lumberliquidators.com/ll/c/x-lft-Maple-Butcher-Block-Countertop-Williamsburg-Butcher-Block-Co.-MABB8/10006983?Wt.ps_id=PS_google_1023&cagpspn=pla&CAWELAID=1925957409&CAAGID=46108513190&CATCI=pla-296303633664&CATARGETID=420003430000674693&cadevice=c&gclid=CjwKCAjw-8nbBRBnEiwAqWt1zQ4qBS_aqbtPIgbYNxIS11YiGpEFXF4JAMjGs8TVoWUUlfQ_VclxhRoCArYQAvD_BwE

Ikea has very affordable counters .

We did granite tile in our first home because Dh really wanted granite and he knows how to tile. We used wood trim, stained to a color similar to the tiles. It kept us in budget and made Dh happy, but I did not love the tile and wouldn’t choose it again. I would look at ikea butcher block and a new sink.

Because of budget, we put RTA cabinets and tile countertops in our starter home, assembling and installing them ourselves, with a bit of help from This Old House (it was an old kitchen, with only a sink on legs and a china hutch for storage). After about a year, I began to regret the tile countertops bigly. Everything stained it, the juice from a cut tomato, tea, coffee, you name it, even though we'd sealed the grout. The only way to get it clean was to bleach it and I shudder to think about the germs that probably lived there.

Seriously, don't do it, and especially don't do it on the existing wood. Tile needs a level, smooth surface. I think I see the sticking point here. If you take that existing wood off, you'd be dismantling the structure that holds the sink and you want to avoid that, right?

Get a carpenter in there to give you ideas on how you can work this. Tile is not a good idea.

Tiled front edges never stay put. Continuous wood or solid surface edges if you must have tile, please.

The budget demands stock laminate and a new to you drop in sink.

It’s always best not to do demo without a clear design plan in mind.

Don't do big fake stone tile. Do tile that is available with the edge pieces--check out Daltile as advised above. Hexagons or 4x4" squares will look correct for your vintage kitchen and tile is a great countertop in spite of the current Rage-Against-The-Tile that is in vogue, especially with the newer grouts that don't get instantly icky. Visit the John Bridge tile forum for advice on all things tile. You WILL need to do something to that surface before tiling.

Also butcherblock is a nice option.

We have granite tile counters and did a strip of granite as the edging. The counter tile hung over 1/4" and the edge strip butts underneath. Not sure that would work with porcelain tile though and would not work with that sink.

There is this metal edging, but again, that would only solve the edge issue and not the sink issue: https://www.homedepot.com/p/Schluter-Rondec-Step-Satin-Anodized-Aluminum-3-8-in-x-8-ft-2-1-2-in-Metal-Tile-Edging-Trim-RS100AE39/202609024

I have installed both laminate and tile counters DIY. I would do laminate all day every day. They are really inexpensive, easier to install, wear better, and are more attractive.

Not on those old cabinets ripe for a gut job. That’s throwing money away.

In the same situation at our rental property in Boston. Was looking at porcelain slabs, but the cost was outrageous. Large format tile w bullnose looks nice, but I am not sure how the edges would be done? Newer grouts are fine for countertops. I found Artistic Tile to be beautiful and it’s doable on the countertops (believe they’re porcelain and do not stain?)

Look at Formica counters. They've come a long way and would be more in keeping with the kitchen you're redoing.

Tiles for counters are not a good idea esp. in the kitchen.

These are Formica brand.

I know this is an old thread, but in case anyone else interested in tile stumbles across it like I did, please don't discouraged by those saying a tile countertop like the one that was originally in this kitchen is impossible to recreate! The trim pieces are available. Many tile manufacturers (expensive and not) make V cap and/or box cap pieces for a countertop edge, although a mud cap piece would work too. And all the sink needs is simple quarter-round bullnose. As vintage-style tile comes back into vogue, there are more options available all the time.

Everything is a matter of personal taste, but I have not found glazed ceramic/porcelain tiles at all hard to keep clean. You just can't let acidic spills sit. Mine are, if anything, easier to care for than the laminate I had before.

I really hope @Lisa Cooperman found a way to keep that beautiful sink!